From what I understand a "slip-up" is the same as a "mistake", is that right?
Hello, JessyA—Yes, a slip-up is a mistake. Normally "slip up" is a phrasal verb. "Slip-up" is the noun related to the phrasal verb.
So, the same way I can use the verb "to make", example: Oops, I made a big mistake in this essay, can I say: "Oops, I made a big "slip-up" in this essay"?
Yes, you can say that, but don't put "slip-up" in quotation marks. It would be more normal to use the phrasal verb: "I slipped up big-time in this essay."
Can I also say, "That's just a minor slip-up, don't worry!", and "I slipped up, sorry. I shouldn't have told her your secret. Sorry, for not keeping my lips sealed"?
Yes, those sentences are fine.