I was using a sinus rinse and something in my right ear popped. What could it be?

Eustachian tube. If the sinus rinse reached the back of your nose (nasopharynx), it is possible that the rinse reached the eustachian tube (channel in the back of your nose that communicates with the middle ear). The rinse may have traversed that tube and caused your ear to pop. Make sure you do not have fluid in your middle ear space.