Algebraic topology is that branch of topology which uses algebraic methods to attach invariants to topological spaces. The underlying philosophy is that spaces can be thought of as being glued up out of simple pieces (simplices, say, or disks, of various dimensions), and so, by studying the kinds of cuts that are needed to decompose a given space back into these sorts of elementary pieces, we can attach invariants to the space. (Think of the definition of the genus of a surface as being the maximal number of cuts along closed curves that one can make without disconnecting the surface.)
In general, to organize the combinatorial information inherent in the gluing of the space out of disks/simplices/etc. into effective invariants, tools from algebra are used. (And so, as an example, the genus of a surface becomes interpreted as being one-half of the free rank of the first homology group of the surface.)
Algebraic topology is not just a significant mathematical subject in its own right, but is also a standard tool in most other areas of geometry, as well as in those other parts of mathematics that employ geometric or topological tools.
Most problems involving algebraic topology involve advanced undergraduate or graduate level mathematics. In particular, a familiarity with the basics of general topology is required, as well as as a familiarity with notions of algebra, especially homological algebra.