A group's base is an area of the board occupied by the group, which provides eye potential for it, and hence a certain amount of stability. It almost always refers to groups on the side and in the corner.
A base is typically created during the opening and early middle-game when it is not economical to spend many moves in one area to make alive groups immediately. A base strikes the balance between stability and development: it is strong, but not too slow.
This two-space extension on the third line is a classic example of a base on the side. Note that this shape is not yet alive, but unless White is very strong in the neighborhood it may not be feasible to attack the black stones effectively.