Player Versus Player

"So this is actually a human player. You can just go into other people's game and fuck them up.
'Cause the game doesn't fuck you enough, it lets other people come in and fuck you up."

—videogamedunkey on Dark Spirits in Dark Souls, Skunk Souls

Player Versus Player (PvP) is a mode of play in Video Games and Tabletop Games where the enemies are other players of the game. This is in contrast to Player Versus Environment (PvE), where the enemies of your character are generated and controlled by a computer's artificial intelligence (for video games) or a Game Master (for tabletop games). The idea of Player vs Player is not only to outplay, outlast, or outwit your opponents, but to put your opponents at as many disadvantages as possible so they can't do the same to you.

Some games have both a PvP mode and a PvE mode. In some cases, the two modes occur at the same time, like in most MMORPGs. A player character can be busy with a quest vs. computer generated monsters in an outdoor zone, and be ambushed by enemy players nearby. This can be either irritating or exhilarating, depending on your mood and tolerance for fellow players, forming the basis of The Great Player-Versus-Player Debate. As a result, many MMO games have PvP and PvE separated, either completely by servers or by areas within the same game space.

Most games have strictly defined modes where only PvE (usually also separating single player and Co-Op Multiplayer from each other) or PvP, but not both, can occur. An exception is strategy games that almost always support an arbitrary mix of human and computer sides, if they have AI at all.

Player vs Player is actually the older of the two modes, since it eliminates the need for any sort of AI.

Not to be confused with the Web Comic PvP.

Examples of Player Versus Player include:

Only PvP

Many very early games. After that, whenever a game's capabilities outgrow AI.

Space War
Pong
VGA Planets
Many engines intended to be generic, rather than sets of much the same with 3-4 different values per entity. At most they can have embedded AI script support (more extensive than merely "mod-friendly" games need) or API for plug-in bots customized for specific rulesets.
- VASSAL Engine "intentionally does not include any artificial intelligence (AI) opponent capability as that is outside its scope". As in, it's "generic tabletop" by design, so AI good even for half of ruleset modules is not feasible.

Primarily PvP

Artillery games such as Scorched Earth, Worms, etc. They usually have AI with adjustable skill level, but it's just not as much fun.
Fighting Games and Racing Games in general have a big focus on players competing against each other.
Mortal Kombat players know that most of its charm is PvP, one big reason the game went online, though you can still play two-player at home.
Team Fortress 2 is nothing but players fighting each other. Even the bots are meant to simulate other players, down to "Virtual Mousepads" that control how fast they turn.
Paranoia was (at least one of) the first Tabletop Games specifically designed to have lots of PvP.
Minecraft has player vs player on by default for servers, though it can be disabled by choice. The feature's uses range from every player for themselves or even clan battles.
Dark Souls has several covenants that cater to PvP action, with different rewards for each.

Separated PvP

World of Warcraft allows players to create characters in different realms. Only in PvP realms can ambushes of that sort happen. In Normal realms you can only be attacked if you have recently attacked other players, certain NPCs, or chosen to "flag" yourself.
- Many other MMORPGs have similar separations.
City of Heroes/City of Villains had several distinct PvP zones. Most were explicitly Heroes vs. Villains, but there was one which was a more typical everyone-out-for-himself free-for-all.
An example that predates video games: The Head of Vecna, using Dungeons & Dragons as the game.