The icon or text which appears next to people's usernames is called "user flair". Each subreddit has its own, and it is set up by the mods.

To get your own, look in the right-hand sidebar for "Show my flair in this subreddit. It looks like: {{ your username }}". It's a little small. If there is an [edit] button next to your username, click it to select your flair. (If there is no edit button, then you can only receive flair from the mods in this subreddit.)