Tabletop Role-playing Games are a large class of commercially-available games. These are usually available only at specialized hobby or game stores, although a few (such as Dungeons & Dragons) can be found in regular bookstores. Game actions are taken primarily through verbal declaration (i.e. "my character climbs the wall"). These are known as "tabletop" RPGs (to distinguish them from live-action roleplaying) or "paper-and-pencil" RPGs (to distinguish them from computer games).
Tabletop Role-playing Games are played sitting around in a comfortable setting (often around a table but not necessarily), and what happens is defined by verbal description. i.e. A player simply declares "I am walking to the window", and it is understood that her character is doing just that. Diagrams and notes may be used as aids, but narration is the primary medium. These are often referred to as "tabletop" RPGs (to distinguish them from "live action" RPGs where the players move around) or "paper-and-pencil" RPGs (to distinguish them from computer games).
There is a wide variety of these games, but they have common features. One person generally acts as the Game Moderator or Game Master (abbreviated GM), who is the authority on the fictional setting (aka "game-world"), and has final say over what happens. A typical game session has the GM prepare a set of challenges for the players in advance. Each player (except the GM) has a single character in the game-world which he controls (known as a player-character or PC). The players then declare what their PC's try to do, and the GM describes what happens.
In practice, much of what happens is either descriptive or obvious results: i.e.
Player: I carefully walk up to the window and look inside. What do I see?
GM: [checks notes] You see a dusty room with a table in the center, which has a wooden box on it. The walls are bare, but there's a door to the inside. Player: Alright, I'm taking my knife and scratching a mark on this window, so the others can identify it later. GM: OK - done. [scribbles this on his notes]
Note that the player uses the first person ("I") to describe what her character is doing. This is just a common way of speaking. Some players use their character's name instead (i.e. "Davidson walks up to the window"). Of course, at other times the results are less clear. For example, the same setup could have more complicated results:
Player: I carefully walk up to the window and look inside.
GM: You see a room with a table in the center. There's a man sitting at the table playing solitaire. He has a revolver sitting on the table beside the cards.
Player: Yikes! I duck down, as quietly as possible. Did it look like he saw me?
This is the basic case of "action resolution" -- ...
It is a bit like cooperative storytelling -- by announcing and describing to the other players what you are doing, you become part of the ongoing story.