Trying to guess what you mean.
A definition of idempotence would be, for $f$ a function on a typical element $x$,
$$f( f(x) ) = f(x)$$
This is an example of a functional equation. So you might be interested in a book or introductory article about functional equations.
There is no limit to the number of properties you can imagine for a function based on a functional equation. Classical algebraic notions such as commutativity, distributivity, associativity can be expressed this way (see below).
Idempotence is just one of the most basic and most meaningful, related to the geometric notion of projection. Even this is too broad to find the kind of elementary treatment you want.
In general a functional equation would be any kind of equality that you can write between a function, its successive iterations and its values. You can complicate matters by introducing derivatives, auxiliary functions, several variables, going into inequations, etc.
As others have already remarked, this is a broad subject touching on very different parts of mathematics. But I would guess you are more interested on discrete objects such as integers or trees than on continuous objects.
You can at first have a look at traditional web sources such as Wikipedia, when you will find a beginning of classification. You can also look at most books or articles about solving problems at Mathematics Olympiad where it is common to have functional equations (on the sets of integers but not only).
Functional equations are used in Computer Science in different contexts such as dynamic programming, automata theory, analysis of algorithms, among others. You may have also heard about Functional Programming. In some of them basic properties of functions are used in evaluating programs and optimizations. For instance, if you can declare that a function is associative, the interpreter or compiler can choose the order in which it will evaluates the arguments since it will not change the result.