Role-playing games into a number of categories, as follows:
For many people the jury is still out on the issue of whether or not card games are a type of role-playing game. In any case the point is probably moot because many role-players also enjoy playing card games (and visa-versa) and role-playing game publishers publish most card games.
Card games are played with packs of special game cards, a player normally starts by purchasing a "starter" pack but can supplement their pack by purchasing "booster" packs. After that cards are exchanged between players either as the result of a trade or by winning them in play.
The cards themselves have various parameters, which are roughly analogous to the suites and face values of a normal card deck. However instead of the suites being Harts, Diamonds and so on, they might be Fire, Water, etc.
The rules of play are just as numerous and complex as the rules for normal card games and so I will not attempt to describe them here, other than to say that level of thought and tactics required are roughly the same as need for Canasta.
Computer role-playing games are very similar to board-type tabletop games, except that even less is left up to the players imagination. Because the computer can represent a scene graphically and move animated characters around it the player only has to worry about the thinking aspects of a game (strategy, tactics, puzzle solving, etc.) rather than the visualization aspects of a game.
LARPs are the great exception to the no acting rule. In a LARP the players dress up in costume and do their best to become actors and actresses.
HOWEVER there are a number of very strict rules for this sort of behavior and these rules are stressed over and over again the rule books for LARP games. Some of these rules are:
PBEMs are very similar to tabletop games except that the medium used for the game is email - rather than sitting around a table talking. As you can imagine the timeframes involved in PBEM games are much longer than for tabletop game: A player must write an email, send it, and wait for the reply to see what happened which might take hours or days, whereas in a tabletop game it would take seconds or minutes.
There are many different variants of PBEMs; generally there is a variant for every type of medium used to host the game. For example:
Honor's Hand is actually a PBDG not a PBEM, I called it a PBEM because that term is better recognized and there isn't too much of a difference.
Tabletop games are so named because they are normally played around a table. Most tabletop games I have played in have been played around a coffee table, but I know people who prefer to sit around a dining table.
Tabletop games can be split into two sub-categories: with board and without board.
Board-type role-playing games are literally those played with a board of some sort (like chess has a chess board). The board usually has some type of grid or maze printed on it, upon which figures can be placed. The idea with board-type role-playing games is that they remove some of the clutter that players would otherwise have to remember, for example: where everyone is in relation to their character, they can simply glance at the board to find out. That said, a lot of the story and details is still left up to the player's imaginations: the sounds of footsteps coming down the corridor towards you, for example is something you just have to imagine.
Board-type games tend to be popular with younger players or very new players. The amount of information that must be tracked during a game can seem intimidating to those not used to it, and board-type games do mitigate this somewhat. However, board-type games are notoriously restrictive: play is restricted to whatever environment is printed on the board. If a dungeon maze is printed on the board then the characters will always be walking around dungeon mazes, if it's an open field then characters will always be outside, etc. Most people (myself included) find board-type games restrictive.
Board-less games are by far the most common type of tabletop game. There is no board, so there are no restrictions placed upon the setting. If the story takes them into an office block of a mega-corporation, or into a cave, or onto a field then the characters are free to go there and interact with whatever they find there.
The storyteller (or game master) must describe the scene to the players and then they can make decisions and choose actions based upon what the characters can "see". Often the storyteller will make quick and impromptu diagrams to help the players visualize a scene.