A roleplaying game (or RPG) is a specific type of game, typically involving four to six participants. Participants are referred to in the general sense as "gamers." White Wolf's roleplaying games, especially those that use the Storyteller or Storytelling Systems, use the term storytelling game to distinguish their games' focus on characters and narrative.
One of the participants in a roleplaying game takes on the role of Game Master, or GM; this title changes depending on the game in question. Other titles for the GM include Dungeon Master (DM), used in Dungeons & Dragons, which serves as the origin for the term; and Storyteller, used in the Storyteller System and the Storytelling System. The role of Game Master is somewhat akin to the "banker" in games such as Monopoly, but the GM also acts as a referee and narrator for the other players.
The other players each create a character using predefined rules. The GM presents the players with a scenario, and the players describe the reactions of their characters. The GM describes the results of their actions, and play proceeds back and forth in this manner.
There are multiple different styles of play, but two of the most common are the "dungeon crawl" and the "storytelling" approach.
Also called hack & slash, this approach is most common with younger gamers and with D&D players, particularly where the two groups intersect. That is not to say that all young gamers or D&D players are hack & slashers, but those two groups display the highest occurrence of the phenomenon.
The main goal of these games is largely the killing of monsters and the acquiring of their treasure. Sometimes there is a reason (acquire a certain artifact, rescue the princess), but these are mostly MacGuffins.
Examples of this play style include descending into a randomly-generated dungeon, and the killing zombie hordes.
This approach is more common with older gamers and with Storytelling System players, especially where the two groups intersect. Again, not all older gamers or ST System players use this approach, but it is most common with them.
Storytelling enthusiasts see their role as less of a "video game" approach and more of a collaborative novel or play. The actions of the characters are less geared towards gaining levels, money or equipment, and more geared towards creating an interesting story — even if it's to the detriment of their characters.
Storytelling enthusiasts tend to look down on hack & slashers as "immature," while dungeoncrawlers see storytellers as "pretentious." Variations on this theme are the majority of sources of conflict between the two groups.