Category theory is the study of dots (objects) and directed arrows (morphisms).
Every morphism has a destination and source object (or domain and codomain); if a morphism f goes from A to B this is usually written f:A->B. Morphisms must have a composition rule: two morphisms f:A->B and g:B->C specify a particular morphism gf:A->C. Composition must be associative and have an identity for each object.
Everything else is just definitions layered on top.
In addition to the familiar categories like Set (the category of sets and functions), there are a lot of cases where morphisms aren't functions at all. One common example is that any ordered set is a category, with the elements of the set forming the objects and a morphism between A and B if and only if A is less than B.
From the perspective of category theory, similarities of structure become more easily evident. My favorite example is the notion of product; cartesian products in Set are exactly the same category-theory construction as greatest lower bounds in an ordered set.