In Latin, do you put the noun that is possessive in the genetive case, or do you do the noun that is being possessed? Or both? I thought it was just the possesser, but in example given in my textbook, copia cibi, copia is not in the genetive, but cibi is. I'm confused!! Thanks.

Only the noun that possesses something is put into the genitive. Yes, just the possessor.

English = John's sister is tall. (Only "John's" would be put into the genitive in Latin.)

English = There are plenty of soldiers in the market place.
(The phrase "of soldiers" would also be put into the genitive because of the use of "of" in this sentence.)

Copia cibi = plenty of food (for men and animals)
The noun after the word "of" in this use is the noun that is to be in the genitive case.