Algebraic topology is the study of intrinsic qualitative aspects of spatial objects (e.g., surfaces, spheres, tori, circles, knots, links, configuration spaces, etc.) that remain invariant under both-directions continuous one-to-one (homeomorphic) transformations. The discipline of algebraic topology is popularly known as "rubber-sheet geometry" and can also be viewed as the study of disconnectivities. Algebraic topology has a great deal of mathematical machinery for studying different kinds of hole structures, and it gets the prefix "algebraic" since many hole structures are represented best by algebraic objects like groups and rings.
Algebraic topology originated with combinatorial topology, but went beyond it probably for the first time in the 1930s when Čech cohomology was developed.
A technical way of saying this is that algebraic topology is concerned with functors from the topological category of groups and homomorphisms. Here, the functors are a kind of filter, and given an "input" space, they spit out something else in return. The returned object (usually a group or ring) is then a representation of the hole structure of the space, in the sense that this algebraic object is a vestige of what the original space was like (i.e., much information is lost, but some sort of "shadow" of the space is retained--just enough of a shadow to understand some aspect of its hole-structure, but no more). The idea is that functors give much simpler objects to deal with. Because spaces by themselves are very complicated, they are unmanageable without looking at particular aspects.