I promised this long ago and am getting to it just now. Sorry, I tend to get distracted and go off on rants. If you have not noticed.
First there is the old childhood game of Let's Pretend. That encompasses things like Cowboys and Indians, House, and whatever else you can think of. Frequently it involves using whatever is at hand to enhance the game. Lines in the dirt to act as house room divisions or a DMZ for war games. Odd pieces of fabric or an old blanket to make a cape for a superhero or a robe for royalty. You get the idea.
There are two basic older (adult) version of this. Historical reenactments is the first. They range from Ren fairs to battles. Form a few hours to all year around. Costumes are elabprate as are props and the general "play area". Various groups and people devote a lot of time and effort into these productions. SCA, DAR, and other groups.
Then there are LARPs (Live action Role Playing) Usually a group dresses up in costume and makeup and act out a game. World of Darkness LARPs were big a few years ago, Shadowrun seem to be now. They tend to be based on table top game ideas and expanded upon. Players usually devote no more than a weekend and many are incredibly creative.
For the players who prefer less public attention but still like face to face social contact there are tabletop games. D20, or any other system involving dice, paper, and a lot of imagination. They are refereed by a GM or DM and can run for years off and on. Genre include everything from Cowboys and Indians (or Zombies) to superheros to good old fashioned Dwarves, Elves and Evil vs Good. Dungeons and Dragons, Starwars, Shadowrun, Cyberpunk; you name it there is or will be a system for it. Or one that can be adapted to it.
Next there is chatroom or even gameroom play via computer. Some sites have die rollers for the game, others just provide a chatroom. Either way, again it is a group getting together to work through improbable problems with basic team efforts.
MUDS, MUSHes, MOOs, MPORGs and MUCKs are next. Frequently these are a game online that can be played with little or no roleplaying, or with full scale characterizations. It depends on the players and the Admins. All have a environment set up through areas built that lend to the atmosphere of the world. Some the IMMs or Admins are very active in setting up a story line others are left open to the player.
PbM - play by mail or play by email. These tend to be be less popular these days, but it is a type of interactive writing between several people. Usually one functions as a gamemaster or coordinator to ensure turns are taken in order and people only see what their character would see. It has been replaced mainly with Play by post.
Play by post or Forum play is very similar to Chatroom play except it allows people to publicly post their character's actions and part of the story when they can instead of having to set an online meeting time. Moderators or Site owners usually function as DMs.
Next time a bit more information on one or more of these styles.