To correct (a mistake) - å rette (en feil) (of Norse origin, remove and address/point out errors and mistakes) or å korrigere (of Latin origin (as is the English "correct"), enhance and adjust). Sometimes interchangeable..
To proofread - å lese korrektur, which is synonymous with å korrigere.
The teacher was correcting essays = Læreren rettet stiler. Which is one instance where using korrigere would sound strange, as the Norwegian (ie Norse origin) word carries all of the pointing out an error and correcting it and -grading-.
I like giving the origins when there are synonyms (similarities?), as our older Norse vocabulary has way more meanings and uses than "recent" imports (all languages share that characteristic, I think). Makes it easier to explain why words might have synonymous use in certain contexts but not others.