Loosely speaking, a loop is a non-associative group. In a group setting, the equation ax = b can be solved for x and by using all of the group axioms, one can show that the unique solution is x = a-1b. In a loop you also know that this equation has a unique solution, but you don't know what it is or how to find it in general.
The simple result which causes our interest in these algebraic forms is: