The field of algebra (sometimes referred to as "abstract" or "modern" algebra) generalizes the idea of an operation like multiplication or addition. The field of topology studies spaces (think "shapes"), but in a less rigid way than geometry. Algebraic topology combines these two, using the tools of algebra to study topology. The study of topology also inspires new algebraic ideas and methods.

In high school geometry, you might have learned about "similarity" and "congruence". These are two ways of considering shapes that are technically distinct as "the same shape". In homotopy theory (the area of algebraic topology I work in), spaces are considered to be the same if you can continuously deform one into another. This leads to a lot of jokes about topologists confusing their coffee cups for their doughnuts and to a lot of interesting math. There's a lot more to homotopy theory than deforming shapes, but the idea of equivalence (how we determine which objects are the same) is a common theme.

If you want to learn a little more about algebraic topology, I have some recommended resources.