Zk proofs are short for zero-knowledge proofs. They are a kind of mathematical system that lets two people prove they know something without either one of them telling the other the thing they know.
Imagine you have a secret number and you want to show someone that you know what it is without telling them what it is. With a zk proof, you can prove that you know the secret without giving away what the secret is. The way this works is you share a special kind of math problem with the other person. They can use the math problem to check that you know the secret, but they don't get to see the secret.