A category abstracts the concept of function composition. You need two things: an identity element, and a composition operation. For functions the identity element is the identity function id(x)=x and the composition operation is function composition compose(f,g) = (x -> f(g(x))). The twist is that you can't compose any two functions; their input and output types have to match up. If g is A -> B and f is B -> C you can compose them to a function A -> C, but the output type of g has to match up with the input type of f. Categories generalise this concept from composing functions to composing any thing with a notion of input and output type. In practice that often ends up being function composition anyway, but with restrictions on the function. For instance if you compose two linear functions you get a linear function, so this forms a category. Or continuous functions, or functions that can be differentiated, etcetera. However, quadratic functions do not form a category, because if you compose a quadratic function with another quadratic function you may get a function with higher degree. For instance if f(x) = x^2 then compose(f,f) is x^4. Polynomials on the other hand do form a category. In general this need not be composition of functions, as long as there's an identity element and a compose operation with a notion of input and output type. Binary relations is the easiest example that I can think of.