A proof is an argument that purports to demonstrate the truth of a given statement, or proposition. If the argument rests upon true premises, it is called logically sound. If it is based on absolutely reliable steps (that is, transformations of the premises using the laws of logic) it is called logically valid.
Ideas about how rigorous a proof must be has varied through history and will still vary quite a bit today depending on the subject matter under consideration. A mathematical proof will necessarily be more rigorous than, say, a "proof" of the correctness of libertarian ideals.
Many arguments for the existence of God take the form of a proof, but all fall short for one reason or another. Some have false premises, others invalid reasoning.