Respawning is the concept of a player reappearing in the game after dying. It also applies to things such as vehicles that have been destroyed and weapons that have been picked up that reappear. Sometimes, especially the first time for it to happen to something, this is called spawning.
Unlike many other games, players can respawn immediately after dying without having to wait a while, unable to play. Along with features like respawning in a convenient location in team-based games and the inclusion of the Hoverboard, Epic intends to give the player "zero time to spectacle". Weapons, ammo, and other items, as well as vehicles are on a countdown timer after they are picked up. More powerful items usually take longer to respawn than weaker items.
See Wikipedia's article for more, general information.
Single player games
In Unreal, respawning doesn't exist story-wise. If the player dies in the single player game, the whole game needs to reset to an earlier state. Killed enemies are permanently removed from the game, and objects that are picked up will never reappear. The multiplayer part does feature respawning, but with no explanation, although respawn points are sometimes marked with a small pedestal.
Unreal 2's similar single player nature means it also doesn't feature respawning, although XMP's respawning is similar to that of Unreal Tournament described below.
Multiplayer games
Due to Unreal Tournament's multiplayer-based gameplay, respawning features heavily. Player spawns are no longer indicated, which makes it a bit harder to spawn camp (wait for other, lightly armed players to respawn, only to immediately kill them). The story of the Liandri Grand Tournament features respawning, but no real detail is given about the way it functions.
UT2003's, and therefore UC's and UT2004's respawning works basically the same as it does in UT. A new feature is that weapon respawn points are indicated with a weapon base. This makes it easier for player, especially ones new to the map, to find weapons, even if they aren't currently available. Ammo and other items don't have bases. Epic invented a new kind of weapon base, called a Weapon Locker for the Onslaught and Assault gametypes. Weapon Lockers hold multiple weapons in a small space, but respawn those weapons for individual players. In other words, if a player uses a Weapon Locker, he won't be able to use it for a while, but other players can still use it, starting their own respawn timer for that Weapon Locker.
UC2's story explains respawning happens through the use of a technology installed in the arenas to spare competitors of actually dying after being killed. Respawning doesn't happen outside of the Tournaments. It is revealed in the final match of the Ascension Rites that the respawners can be disabled at will in each arena.
UT3 explains that Respawners, originally in the Tournaments, are now used in real war to keep soldiers alive and thus have less casualties. However, some portable Respawners need Field Lattice Generators (flags) to operate, so they can be disabled by capturing flags and decreasing the energy of the enemy Respawners until they turn off. In deathmatch-type battles, Respawners have a non-renewable energy source that are used only when Respawners are working (respawning people), so they can be disabled simply by killing enemy team members continuously until the enemy power turns off.