It is a diagram whose vertices are mathematical objects and whose arrows are morphisms (often functions) between those objects such that any two paths between t

It is a diagram whose vertices are mathematical objects and whose arrows are morphisms (often functions) between those objects such that any two paths between two vertices represent the same morphism/function. Typically it is used to infer information about one morphism or object using others, by keeping track of many properties at once in a graphical fashion.