Grammarians: What Is The proper name for this Word Ending?

I’m talking about the endings of nouns: “-er” AND -“EE”; AS IN “EMPLOYER” AND EMPLOYEE". It seems a rarely used construction in English-we don’t commonly use it in nouns like “eat” (eater and eatee?).
Anyway, is there a name for this? Where did it come from?

Noun Suffix.

Because it’s common for people to be either an employer or an employee. Perhaps if we lived in a cannibalistic society, “eater” and “eatee” would be common.

-er is an agent suffix.
I’m not sure what -ee is.

-ee can be either an agent suffix or a beneficiary suffix.