Instances are special areas that are entirely seperate from the "main" game world, that are created for you and your group (along with an enemy group in the case of RvR scenarios) in the instant that you start a new scenario or instanced quest. You can think of them as alternate dimensions, one dimension for each instance, meaning that any number of people can be exploring the "same" area but never see each other. Each instance will have its own copy of everything, including monsters, loot, etc.
There is minimal instancing in dungeons. Dungeon instancing is generally restricted to final boss encounters.
Originally most, if not all, of city sieging was intended for instancing. However, this has been changed due to play testing. Exactly how much remains instanced is currently unknown.
See also Wikipedia:Instance dungeon.