A branch of discrete mathematics. The study of relations, connections, networks, connectedness, and especially graphs. A graph is a pair of sets. The elements of the first set are called vertices. They can represent people, words, letters, cities, brain cells, web pages, countries, or anything that can be associated with, or connected in some concrete or abstract way, to something else. The elements of the second set are pairs of elements from the first set. They are called edges. They can be unordered pairs or ordered pairs. In the latter case, the edges are called directed edges, and the graph is called a digraph.
To picture a graph, think of dots connected by line segments to other dots. Each line segment must have a dot at both ends. Neither the shape nor the length of a line segment matters in graph theory. All that matters is which dots are connected to which dots. You can give them names, numbers, letters, and/or colors. Or you can color the edges.
Graph theory can help you to understand games, hierarchies, networks, family trees, food webs, flow charts, algorithms, trains of thought, just about anything you can think of!