An Instance is a level or map that only some people can enter. It may be PvP or PvE based. Typically, a particular instance may be a 10 vs 10 match, or a small squad versus the AI opponents. Also, the Base Raid mechanic allows for larger battles of 60 vs 60 to take place across several instances of the same map. Once, for example, a group enters an instance, they will usually be at least partially cut off from the game world they were in before they entered.
The reason why massively multiplayer online games (MMO´s) use instanced missions or levels is simple: to maximize performance of the game client. For example, if 10 players enter an instanced mission together, they will only have to send data to eachother and the main server during the mission, as compared to potentially hundreds of other players. This reduces bandwith and allows a smoother game experience. The amount of instancing used in MMO´s varies. Multiplayer games that do not rely on turn- or timer-based game mechanics (ie, a MMORPG-FPS like Global Agenda) usually use more instancing, because they require more bandwith and the game experience suffers more with higher connection latencies.