Given a ring $A$, an element $e \in A$ is called an idempotent if one has $e^2 = e$. If $e$ is an idempotent, then so is $1 - e$, since$$(1 - e)^2 = 1 - 2e + e^2 = 1 - 2e + e = 1 - e.$$Also, we have $e(1 - e) = 0$. This is a special case of the following situation.
A collection of elements $e_1, \dots, e_n \in A$ is said to be a set of orthogonal idempotents if one has$$e_i^2 = e_i \text{ and }e_ie_j = 0 \text{ for }i \neq j.$$
My question is, what is the underlying intuition behind idempotents and orthogonal idempotents? I am finding them quite hard to work with...