"Correct" and "right" are essentially synonyms when used in a certain situation or context. So, for example, if someone asks the following question: "You are a boy with black hair?" I can say, "That is correct" or "That is right." They both mean the same thing. However, if someone says, "Which direction should I go?" And I say "Go to the right" Then it is different, because it is referring to a direction. For example, I have a right arm and a left arm, but I can also be right (correct). I hope that makes sense. Let me know if you have any questions.

I will add upon what William has said (all of which is true). If you were to say, "it's the right thing to do", that could be taken two different ways, depending upon the context. It could mean the same thing as, "it's the correct thing to do." "Right", as William said, can mean basically the same thing as "correct", such as, "no, that's not how you cut the cake. Let me show you how to do it the right way." However, the sentence, "it's the right thing to do" could also be talking about morals, as in, "I did the right thing and told him the truth," or "my brother ate my last cookie, but I didn't kill him because I knew forgiving him was the right thing to do." "Right" in this case refers to something that is morally good. Hope that helps. :)