It almost seems as though a combinatorial proof is "Explain the intuition behind this relationship (using normal words) to explain why it is true." 

I'm a little lost as to how this is a proof exactly, since I always thought proofs were meant to be very rigorous. I can't think of a counter-example but it seems possible to come up with a combinatorial proof that sounds right but is actually wrong when you get down to the algebra.

Is my understanding correct? Are there any other conditions that a combinatorial proof must meet to be considered a valid proof?