So, sometimes we want to show that something is true. This is called a "proof". Kind of like when you show your teacher that you did your homework and got all the answers right. You're proving that you know how to do the math problems!
But sometimes, other people might not agree with your proof. They might say, "Hey! That's not right! You're missing something!" This is called a "refutation". It's like when your friends say they don't believe you can do a backflip, so you have to show them that you can.
Now, when people are talking about "proofs and refutations" in general, they usually mean something more complicated. They mean that people are trying to prove or disprove really big and complicated ideas, like "All humans have rights" or "There are other planets with life on them".
These kinds of ideas are really hard to prove or disprove completely. It's not like math problems where there's just one right answer. Sometimes people can come up with good arguments for something, but then someone else comes along and says, "What about this thing you didn't consider?" and then the first person has to think about it some more.
This back-and-forth of trying to prove something, and then someone else trying to refute it, can go on for a really long time! People might present different proofs or refutations to try and convince others of their ideas. And sometimes, even after a lot of arguing, there's still no clear answer!