Endorser: the person who wrote on the back of the check
Endorsee: the person the check was endorsed to
Now granted, in order to deposit the check, the endorsee will then need to endorse the check, which will make them become an endorser, so it creates a circle of the payee becoming the endorser, who creates a endorsee, who then becomes an endorser and a new endorsee is created (the bank, technically), and the bank then becomes an endorser, etc., but that's the joys of this s—, uhh, stuff.
Generally, “-er” is the person doing the actually, and “-ee” is the person receiving the action. So, the payer of a check is the person who is paying it. The payee is the person receiving it or that it is being paid to. The word endings work the same way here. Since endorser and endorsee are such usual words, it was easier for me to remember by thinking of other words with “-er” and “-ee” endings. Not sure if “payer” and “payee” would work for others, but if you can think of any “-er” and “-ee” words you're familiar with that are used the same way, maybe they'll help. 🙂