Roleplaying games come in many forms. First, there is a whole category of sexual activities that are also called "roleplaying." That's not the topic of this post. There are also two categories of video games called roleplaying - cRPGs such as Diablo/Diablo 2, Neverwinter Nights, Elder Scrolls, etc., and MMORPGs such as World of Warcraft, Guild Wars, Everquest, etc. Then there are the pen-and-paper style roleplaying games. In these, two or more people get together and create a story. One person generally controls the setting and plot, while everyone else controls a character (or several).
There are a multitude of ways that a group of people can get together to create a story. Tabletop is where people gather around a table, usually in someone's living room, and there is usually dice involved. LARP (live-action roleplaying) is like a combination masquerade ball and improv acting event - participants dress up as their characters and interact with each other as them. Sometimes there is a strong storyline, and the improv is mediated by a narrator or group of narrators; other times it's an excuse to pound on each other with foam weapons. People also sometimes play by phone or through the mail. With the advent of the Internet, there came email and Skype, and there are several ways to play online.
Chat-based games happen through Instant Messengers like AIM, MSN, Yahoo, or through IRC. There is also specialized software dedicated to them, such as OpenRPG. It's a good realtime substitute for when the group can't all meet around a table, it's slightly slower than tabletop, and it's limited mainly by participant typing speed.
Then there are the true online writing games. They take place through blogs, wikis, and forums. They run much slower than chat; there is no instant gratification of seeing immediate feedback from other participants. The tradeoff is that the writing can become very well-thought-out and elaborate. They range from games that are just like tabletop in feel, up to full-fledged collaborative writing.
The hobby is about 40 years old at this point. For some reason it's not well regarded, even among other geeks. I personally see nothing embarrassing about it, nor reason to avoid mentioning it in non-gaming social contexts. Roleplaying is group storytelling, and storytelling is as old as humankind.