Respawning refers to a creature starting over after being killed.
For player characters (PCs), respawning is a way to continue playing after death without resorting to loading a saved game and without relying on someone casting raise dead or resurrection. Whether or not a PC can respawn is determined by the module being played, as is whether the PC continues at the place of death or is sent to a special location upon respawning. Also dependent on the module is the size of the penalty for respawning (if there is one).
In the official campaigns, the penalty for respawning is 50 experience points per character level and 10% of the PC's gold pieces, unless those penalties exceed certain maximums. The maximum experience penalty is 1 less than the amount that would cause the PC to lose a character level, and the maximum gold penalty is 10,000.
BioWare allows respawning throughout most of the official campaigns. In the original campaign, respawning is only forbidden in a few special areas. In the Shadows of Undrentide (SoU) campaign, respawning is forbidden in the interlude and chapter 2. In the Hordes of the Underdark (HotU) campaign, respawning is forbidden in chapter 3. Furthermore, a PC is required to have a certain item in his inventory in order to respawn (when otherwise allowed) in the expansion campaigns, namely focus crystals (in SoU) and rogue stones (in HotU); one of these items is consumed with each respawn, and if the PC runs out, respawning is no longer an option.
When applied to non-player characters and monsters, respawning can refer to a creature returning to its starting location fully healed after death. However, the term can also refer to a random encounter creating creatures after the previously-created creatures were all killed. In the latter case, the respawned creatures might not match the dead creatures exactly since the randomness of the encounter comes into play again.