Most keyboard that you find on stores nowadays (basically all of them except the high-end gamer products) use a silicone membrane with a silicon dome under each switch, having each dome a small contact that will trigger the keypress once you push the key down. After some time of usage the contact gets wasted and sometimes you have to push the key down harder to make a keypress.

Differently, mechanical keyboards use actual, physical switches underneath the keys to determine the user has pushed a key. When you press a key, you press it's switch down. When you press the switch down, the keyboard sends a signal to the PC telling it that you pressed that key. These switches are far more durable than the rubber domes and have two very important features: a 'click' sound that provides direct feedback to your ears and a mechanism that automatically pushes the switch down once you push it to a certain point. Unlike rubber dome keboards, where you just know that the key has been pressed when you look at the screen, with mechanical keyboards you have tactile and audible feedback that let you know that the switch has been pressed for sure. There are some claims that due to this direct feedback, you can type 30% faster with mechanical keyboards. There are also other claims that say that you get less 'finger fatigue' with mechanical keyboards because the mechanical switches do part of the pushing down force for you, but you probably only need to worry about this if you're a programmer or professional gamer.

Among mechanical keyboards you can find several types of mechanical switches, being the most common nowadays the Cherry MX switches. You can find them in several variants, having each a different type of audible and tactile feedback. These are the switches you find in expensive high end gaming products that are available in the stores. Since they are still manufactured you can buy individual switches and keycaps in several online stores.

Another type of switch that was very common was the ALPS and the SMK Alps mount switches. You can find these in many old keyboards. Some of the keyboards that I know that have them are the Siig Minitouch, the Monterey K108, and the Focus-2001. I'm sure there are many more, but these are just the ones I know of at the moment. I actually like more the SMK Alps mount switches than the Cherry switches. However, unless you find them on old keyboards I never found any store that sells the individual switches nor keycaps for these kind of switches.

There are also other variants of mechanical switches like the Topre, the buckling springs, and others, but I'm sure you can find more information in the links below if you're interested in this topic :) .

More information here:

http://www.pcworld.com/article/242037/mechanical_keyboard_faq_pick_the_right_switch.html

http://www.pcworld.com/article/240939/mechanical_keyboards_should_you_switch_.html

http://lifehacker.com/how-to-choose-the-best-mechanical-keyboard-and-why-you-511140347

http://deskthority.net/

https://geekhack.org/