Category theory is a type of mathematics. Category theorists show how different ideas in mathematics are alike. For example, some ideas from topology and abstract algebra are similar. Ideas in category theory are written down in formulas or diagrams. Category theory can be used to make computer programs more secure or easy to write.
A category is a mathematical object. It can be drawn with dots and arrows. Each arrow goes from one dot to one other dot. There can be many dots, sometimes so many that you cannot count them. An arrow is the abstract idea of a function and the dots are the domain and codomain of the function.
Composition is a way to make a new arrow out of two arrows. If two arrows form an elbow shape, then they have a composition arrow from the start of the first to the end of the second. Composition obeys some axioms, or equations.
Many ideas in abstract algebra are examples of categories or can be used to make categories. Often, this makes the ideas seem more similar.