Intuitionistic logic is a logic that is weaker than classical logic in the sense that there are some proofs in classical logic that don't work in intuitinistic logic. Specifically, the law of the excluded middle (either A is true or not A is true") does not hold.
You can convince yourself that this is a reasonable type of logic by interpreting "A is true" to mean "I have a proof of A", and likewise "not A is true" means "I have a proof of not A". Then it's pretty reasonable that you neither have a proof of A or of not A.
I haven't read Godel's proof, but my understanding is that he uses intuitionistic logic in some of the key steps. Since facts in intuitionistic logic are also facts in classical logic, then I believe the incompleteness theorem holds in both logics. But I'm not at all an expert here, so someone please correct me if I'm wrong.
"Intuitionism" also refers to a specific philosophy of mathematics. But a lot of people just use the word to refer to the logic that came out of that philosophical school.
Intuitionistic logic is also very common in type theory, so a lot of proof assistants use it.