Correct spelling, explanation: prove comes from the Latin word probare. It may seem weird, but it’s good to know that in linguistics, b often is changed to v, which is the process that happened here. Then, of course, the word lost the Latin suffix -are, and instead changed it to an easily modifiable ending e. And if you wonder what about proof – it stems from the same word as prove, but it has undergone much more changes due to being used more. Try to keep in mind not to confuse these two!

Definition of prove:
verb
1. to give evidence of truth or untruth
The good ratings of the book prove that the author writes well.
Unknowingly he was at the crime scene, and now he has to prove he’s innocent.

Idioms with prove:
have something to prove – to feel the need to show one’s abilities to a person who doubts them
The movie did too many special effects; it looks like the special effects team has something to prove and they outdid themselves where it wasn’t necessary.
prove oneself as something – to show one’s capabilities in a given field
She was not the most hard-working student, but I have to say, with this blooming business she has proven herself as a good entrepreneur.

Incorrect spelling, explanation: seeing that proove is incorrect, and yet proof is okay may be confusing. The truth is, both words come from the Latin probare, and its later French form – preove. The main difference is that while proof has been used more often and in different contexts, it strayed a little further from the source. Prove, on the other hand, is still quite similar to the French original word.