A proof starts with a set of known facts, then derives a conclusion that must be true given those known facts. From the perspective of professional mathematics, the key feature is rigor -- the conclusion must be true.
Problems with Proof in Modern Mathematics
There are problems with this idea of proof. The first is, what known facts can you use in a proof? The second is assessing rigor -- mathematicians themselves cannot tell if a proof is rigorous. To be more precise, they might notice a problem with the rigor, but if they don't notice a problem, it could be that the proof is rigorous, or it could be that it has a problem but it wasn't noticed.
A third issue came up with the computer program that proved the four-color problem. It in a sense provided no insight into why the proved conclusion was true.
Using the Concept of a Proof with Elementary Students
For elementary students, it might be impossible to teach the concept of proof. I don't know. I have had no desire to introduce the concept to my third-graders, because I think it would fail completely. But I have started introducing the word, if not the full concept, to my sixth graders.
But for teaching purposes, at least for elementary students, we need to change the concept. A "proof" is an explanation of why something is true, based on simpler facts. For example, you might prove that the area of a right angle is b*h/2, using the known formula for the area of a rectangle. You might then prove that the area of any triangle is b*h/2, using the known formula for the area of a right triangle (or a rectangle again).
The emphasis here is explanation -- showing how something new follows from something we already know. Put another way, the goal is understanding -- understanding why something is true. As a teacher, you can think also think mnemonic. When a student understands why some new fact is true, the student will remember the new fact.
So, when you ask a student to prove it, that should be a call for rigor in the student's explanation. But moreso it should simply be a request to understand and explain why it is true.