I think I’ve figured out most parts of the cubical type theory paper(s); I’m going to take a shot to explain it informally in the format of Q&As. I prefer using syntax or terminologies that fit better rather than the more standard ones.
Q: What is cubical type theory? A: It’s a type theory giving homotopy type theory its computational meaning.
Q: What is homotopy type theory then? A: It’s traditional type theory (which refers to Martin-Löf type theory in this Q&A) augmented with higher inductive types and the univalence axiom.
Q: So what are HIT and UA? A: A HIT is a type equipped with custom equality, as an example, you can write a type similar to the natural numbers but with an equality between all even numbers. Well, it cannot be called natural numbers then.