"A reasoning conducted according to certain rules in order to demonstrate some proposition (statement, theorem); it is based on initial statements (axioms). In practice, however, it may also be based on previously demonstrated propositions. Any proof is relative, since it is based on certain unprovable assumptions. Rules of conducting a reasoning and methods of proof form a main topic in logic."
-Proof. A.S. Kuzichev (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Proof&oldid=18441
For more information on what proofs are and how to do them, please the basic guide here as well as the textbooks listed below, held byt YOUR AUM library.