There are things in life that I don't understand. The rules of rugby. The continuing success of David Guetta. How to do an overhead kick on Fifa 12.

"You press Y and A really fast, like almost at the same time," says my son Patrick, who's six. I watch as his small thumbs flip between buttons. He could play computer games before he could read. Now he reaches for his Nintendo DS like I reach for my mobile; he fills in idle moments on Fifa, playing games or altering his team or practising shots. I don't mind, except when he gets so wound up by a vital match that he cries. My emails don't make me do that.

I go to grown-up events – weddings, anniversaries, all-day lunches – and each time, at some point, I see young children gathered around a device: a phone, an iPad, a hand-held console. They're absorbed, and quiet, not ruining anyone's day, which is a good thing. Isn't it? After all, when it comes to kids, there's not much point in pretending technology doesn't exist. It would be like pretending Lego didn't exist. When boys go round to each other's houses, they play football (in real life), or they play football (on X-Box 360 or PlayStation 3) – or they jump on top of each other in a big bundle and roll around and yell.

If I'm honest, my son – and even his sister, who's one and a half – have an ease around technology that I find scary sometimes. The baby scares me because she keeps deleting stuff off my iPad. Patrick scares me because he could use the Nintendo Wii controls, shift from game to game, choose players, set up teams by the time he was four. He still can't tie his shoelaces. There is research that says he is not alone: a survey of 2,200 mothers in 11 countries found that 70% of their two- to five-year-olds were comfortable playing computer games, but only 11% could pass the shoelace test.

Most kids' shoes have Velcro straps, of course. The shoelace thing is fine. But computer games still bother me. It's the knowledge gap. I have no idea what Patrick's up to when he plays Zelda, or cries over penalties in Classics XI, because, other than the odd game of Space Invaders, I've never got into computer games. Mostly, what my kids play with is a variation of something I had myself when I was young, so if they get stuck, I can help. But with computer games, I am as useful as an instruction manual for a Commodore Pet.

Michael Acton Smith is a genial man aged 37. He has big hair and wears black clothes, which means that business journalists describe him as rock 'n' roll. Actually, he's nerdier than that: more of a non-stop enthusiast, a man as dedicated to his weekly mates-together football game as he is to his business. That business? Oh, just Moshi Monsters.

Moshi Monsters is a UK-based website for young kids, where they pick their own monster, customise it and take it exploring: off meeting other Moshis, playing puzzles, earning points, decorating its home, acquiring cute pets called Moshlings by growing flowers… Sounds dreadful? You're clearly not aged between six and 12. Half of British children that age have, or used to have, a Moshi pet. Worldwide, there are 60m users and rising; one child every second signs up to the site. It only came online in 2008.

Gone are the days when children sat rapt in front of TV programmes, consumed films or books passively, bought the merchandise and that was that. Kids want input – they've always drawn alternative characters, dressed up as their own superheroes – and what the digital world does is give them that opportunity. It lets them shape and share their entertainment. So, with Moshi Monsters, not only do they customise their avatar-monster, but they get to design its room, spend their Rox (Moshi currency, earned by completing tasks), choose their Moshlings. And they can communicate with each other.

All of which brings brilliant, inspiring excitement for kids; and instant, hum-dinging, digital Fear for their parents. Michael understands this. (Actually, I think much of what he does is about balancing children's desires with that of their parents.) He calls Moshi Monsters a "walled garden", a training environment, and points out that communication between users is very carefully monitored. You can't use the social aspect, for instance, unless you (the parent) sign up with an email address, from which Moshi obtains and stores your IP address.

But he really believes that Moshi's small social networking function is useful for children. His argument is: you don't want to let your 13-year-old loose in the brutal social boxing ring of Facebook without some kind of primary-school fitness training.

"We wanted to create a middle ground," he says. "Something between an internet Wild West and a site that is completely locked down. We treat kids with respect, we know they are smart, and it's up to us and their parents to give them guidance about how to behave in this brave new internet world. It's the same as giving them guidance on how not to talk to strangers and how to cross the road. With Moshi, they learn in an environment that is deliberately designed for them."

Even in real life, Moshi Monsters is quite an environment. Its HQ is housed in the Tea Building in east London. Its reception resembles a Tahiti beach bar. There are Moshi Monsters jelly-beans for guests; also table football and an old (working) Pac-Man arcade game. It's like a Hollywood set designer's idea of the inside of Nathan Barley's head. Still. Moshi isn't big among my son's friends, I think, peevishly. Not the boys, anyway. The site's bright colours and big-eyed characters are too cutesy, the games not competitive enough.

I tell Michael about my son and his crew of Moshi refuseniks. He is as unfazed as a man with a thriving £125m business should be.

"Girls click with Moshi a little more than boys," he says. We're sitting in a small glass meeting area, off the main room. "The split is about 65% girls, 35% boys. Boys drift away once they get to around eight or nine. When they're there, they're more interested in the competitive elements of Moshi – the puzzles, the games. The idea of mastery is much more of a boy thing: that's why they absorb all the data on Pokémon cards, or learn all the flags of the world. Girls love the social aspect."

And it was when Moshi Monsters introduced that social element, in 2009, that the site really took off. "Many kids don't want to just play online; they want to experience stuff with their friends, share gifts with them, show off to them. Similar to grown-ups," says Michael. "And we don't want them uploading photos or giving location details, and we don't allow them to private-message, but I think it's wonderful that they can send a virtual gift to their friend, or leave short messages on their pin board, or chat in forums."

The most common social problem Moshi users have is with kids sharing their password with a school friend and then that "friend" going into their room and cleaning it out. What Moshi does in such a case is try to get the person to give the stuff back; or, if there's been some bullying, then they can ban a user for a short time – even permanently. Not only does the Rox-thief learn a lesson but the innocent party begins to understand how to behave safely on a social network.

Recently, Moshi Monsters expanded out of the virtual into the real world, with a magazine, which sells 200,000 copies a month; also trading cards, soft toys, Lego, stickers, bobble bots (don't ask), books and music. I have to say that I loathe all this tat, despite Michael's efforts to keep everything Moshi of good quality, stimulating and suitable for young consumers. But then, he doesn't have kids. He doesn't have to tidy up such quality tat every single day.

Before I leave, I ask Michael why he thinks Moshi Monsters has succeeded. He tells me it's because of three elements: nurture, social and story. Nurture: kids like to care for things, from a doll, or a real-life pet, to a virtual pet, or imaginary friend. Social: they like to interact with each other. Story: they get bored quickly. If an online environment doesn't have a decent story embedded deep within its psyche, then kids will just stop using it.

I come away from Moshi enthused. I like the site's quirkiness – how the six core Monsters were just made up from silly sketches, without market research. I also come away with a game for Nintendo DS, called Moshling Zoo. This was Number One in the DS charts for 15 weeks, the longest-ever chart topper.

I give it to Patrick, who plays for around five minutes. "It's just searching for stuff, Mum," he says. "And then finding it." What he's into, tech-wise, is games: playing tennis or cycling on the Wii, Angry Birds on my iPad, Fifa 12 (football) or Zelda (a puzzle adventure game) on his DS. Often, he re-enacts the highlights in real life, so we can appreciate his genius skills.

Is any of this beneficial? I have no idea. He really enjoys playing his games, but he also really enjoys stuffing his face with white sugar lumps and I don't encourage that. So I talk to Dr Sonia Livingstone, an LSE professor who's on the executive board of the UK's Council for Internet Safety and directs EU Kids Online. According to Dr Livingstone, there are two common areas of worry for parents. First: how long their child spends playing games. Second: the content of those games.

"Parents focus on the fact that the game involves football, or killing. But the point for the children is the skill levels, competition, gaining points, beating others. It's not the content, not really. Children have a lot of skills they have to master: like reading and writing, maths, eating with a knife and fork. With games there are skills they choose to master."

But are those skills useful? I know that chess stimulates kids' brains and Swingball improves their hand-eye co-ordination, but does playing Super Mario Kart do anything for my son other than keep him quiet for a bit?

Dr Livingstone explains that games where one level leads on to the next challenge can be good for reasoning. Repetitive games that you play by yourself are less useful, though they can help with persistence and concentration. Though she's careful not to recommend anything in particular she mentions that, for kids older than my son, virtual world games such as SIM, SIM City, or World of Warcraft (a roleplay game) can be creatively stimulating.

Her advice for parents is two-fold. First, get online with your children when they're little, to help them figure out how to manage their internet use. "And second," she says, "less shouting at the kid to get off the computer and more, 'Let's go to the park.' We're a busy bunch of parents and it's easy to put them in front of a game while we get on with stuff we need to do. All kids say computer use is a time filler – a way of avoiding boredom."

Boredom. My childhood was filled with it. Hours spent watching raindrops crawling down the window. Days doodling alternative worlds, or inventing radio shows on play-and-record tape machines. Today's kids have far more tedium-killers, from all-day cartoons to a never-ending parade of squeaking, beeping, speaking electronic toys. Plus, they expect us to help them with their entertainment. I don't remember getting instructions from adults about how to do stuff, other than how to ride a bike. And even that was just a sharp push in the back as you careered downhill towards a ditch. Generally, we were left alone to get on with whatever we were doing.

These days, not only are we encouraged to show our love for our kids but increasing awareness of children's rights means we also have to take more care of them. The law insists. You can't drive without them being safely strapped in. You can't leave them alone while you nip to the shops. No way would you chuck an eight-year-old out of the house at 9am on a Saturday and tell him not to come back before tea.

Perhaps it's no surprise that we occasionally shove them in front of computer games. Yes, it's to give us a break from them. But it's also to give them a break from us.

Dr Livingstone tells me that British attitudes to technology differ from other European countries. Essentially, Britain, the Netherlands and Scandinavia are well ahead compared to other places in Europe when it comes to digital life. But the British are by far the most worried about the internet – we're the most worried about everything. In the Nordic countries child rights are quite advanced, and they have confidence in the good sense and safety of their children. In southern Europe, parents are more confident in their position as heads of the family. They don't feel guilty if they tell their kids to get off the net. Also, the weather means children mostly play outside anyway.

In short, in the UK the weather is bad and we're worried about paedophiles, so kids are stuck inside staring at screens. Luckily, paedophiles don't know how to use the net.

I can remember the first time I saw a computer game. It was Pong, ye olde tennis game: slow and unsophisticated, as thrilling as playing ping pong with a balled-up sock. No wonder I never became a computer nerd. But what if I'd been born later, spent my teens in a time where computer games were as commonplace and fun as Snakes and Ladders? Would I be as excited about introducing my children to classic video games as I am when I read them my favourite Roald Dahls?

I talk to Dan du Preez, a 28-year-old computer programmer who lives in Leeds. His son is six, and he's into Minecraft. This is an online world that looks a bit like Lego: everything in it is made up of blocks. It's massively popular: 9m games have been sold so far, and it's only been out in its full version for 18 months. In Minecraft, there's no end point; you just build things, create your own environment. Though, to spice things up, there's an attack mode, where spiders zoom in to destroy what you've been building: you can switch this on and off.

Dan is confident around technology. When he was young he enjoyed chat rooms, but this doesn't mean he wants his son using social networks – "not before he's mature enough to know how to behave on there, and have a thick enough skin". And he's not keen on multi-player online games, either for himself or his son, because you have to be at your computer at a certain time in order to play, and "you can't pause the action, which kids just don't get".

What he is keen on is children learning what computers actually are. He's a fan of Code Club, which aims to teach schoolchildren how to programme computers, and thinks it was better when consoles were simpler, not so beautiful. You don't want to take the back off an iPad and fiddle with its insides. It's too perfect, and far too expensive; whereas the computers that Dan grew up with were clunky and basic, easy to take apart and understand. When his son is a little older, he plans on showing him that there is more to an X-Box than just playing the games. Dan will reveal – ta-da! – that there's a computer inside the handheld console and that his son can make it work differently.

This is so far beyond me that I feel rather downcast. Oh well. Apparently, you can build a virtual computer on Minecraft, so perhaps Patrick and I can have a go at that.

Dan says: "When I worked in a games shop, I saw a lot of parents buying games that were totally unsuitable for kids, and I'd point it out, but they wouldn't take it on board. The best way to find out what's OK is to rent games and play them yourself." Fine for him to say. I don't think Dan has ever spent two hours trying to get Lara Croft to stop walking into walls.

Moshi Monsters are nice enough to let me bring some children into their workplace, so I take Violet, who loves Moshi, Francesca, who likes the cards and the merchandise and Patrick, who still doesn't much like any of it, but politely tries a couple of puzzles before we go.

At Moshi HQ, they ping about like pin-balls, utterly thrilled by their environment: the colouring-in wall, the arcade games, the Tree House meeting room. But by far the most exciting aspect of the visit for them is the gum-ball machine. They chew gum throughout; neon saliva drips from the corners of their mouths. At the end of their visit, they are brought into a room to try out some carefully presented merchandise. All three ignore everything in favour of hurling themselves on a pile of beanbags and screeching.

But then, that's kids for you. They have their own way of doing things. They're always fascinated by something off to the side, they focus in on stuff you never really noticed. You can take them to a playground, but you don't know if it will be the slide or the rope swing or the little switch on the water fountain that grabs their attention.

Even if I played every computer game that my kids do, what I enjoy about the experience is not what they enjoy. I like the silliness of the graphics, the funny noises. They take the medal positions deadly seriously. It's their party, not mine. All I can do is visit: check the fire exits, make sure everyone else there is the same age and ensure they don't eat so many sweets that they puke on the way home. Oh, and ensure that a responsible adult – me? – is around, for emergencies.