Direct proof: "P is true, see?"
Proof by construction: "P cannot be false because here is a case where it is true."
Proof by contradiction: "Q is known to be false. If P is true, then Q is true. Therefore, P is not true."
Proof by exhaustion: "All cases of P can be formed as a case Of X or Y. X and Y are both true. Therefore, P is true."
Proof by induction: "P is true in the case N = 1. P is true in any case N+1 following case N. Therefore, P is true for all cases where N is an element of the set if positive integers."
Proof by handwave: "P is true because it is. Happy? No? Tough."