Combinatorics studies problems involving finite sets of objects that are defined by certain specified properties. For example, the objects in question may themselves be sets, numbers, graphs or other geometrical configurations.
Enumerative combinatorics is concerned with counting the number of objects of a certain kind.
Extremal combinatorics is concerned with finding the optimal objects of a certain kind.
Topological methods, algebraic methods and even probabilistic methods have been used to solve combinatorial problems.
Computer algorithms have also been used to solve some seemingly intractable combinatorial problems.
Conversely, combinatorial methods have been used successfully to solve problems in many areas of mathematics and computer science.
Here is a sample problem that would use combinatorics:
Strangers and Acquaintances (F.P. Ramsey 1930):
What is the least number of people that you need to have in a room so that there is always a group of three mutual strangers or a group of three mutual acquaintances? The answer is six.
Combinatorial game theory is a branch of mathematics devoted to studying the optimal strategy in perfect-information games with two or more players (typical), one player (puzzles), or zero players (like Conway's Game of Life).
Combinatorial games probably fit elsewhere, but software that actually applies combinatorial game theory is appropriate.