A conjugate is a sequence of the form X.Y.X-1, where X-1 is the inverse (or backwards operation) of X.
For example, suppose Y is an operation that cycles 3 corners on the top layer:
Then it is also possible to use Y to solve the following cube:
This can be achieved by rotating the blue face so that the bottom corner moves up to the top layer so that all three corners are in the right position to apply Y. Then apply Y. And finally, rotate the blue face back to where it was at the very start. If we call rotating the blue face operation X, then this whole procedure has the form X.Y.X-1.