Dimensions are a way of separating parts of the game world from each other. Each dimension can contain any element, and elements are only visible to players in the same dimension as they are. You can in theory have up to 65535 dimensions.
A useful use of this is for interiors. For example, there is only one actual Pizza restaurant in San Andreas, with each warp point going to the same place. This works fine in single player games (as you aren't going to be in two places at once), but with multiplayer this can be confusing, as you could enter in one place and see all the players who entered in another. To avoid this, you can split the players into dimensions, so they will only see players who entered in the same place.