When is Eurer used?
It occurs relatively rarely, but if you translate "I am giving meat to your (plural, informal your!) cat", for example, you would see it: "Ich gebe eurer Katze Fleisch".
The "er" ending is because the case is dative (Katze is the indirect object) and the following noun (here "Katze") is feminine.
dative form of female singular possessive pronoun with a group of "possessors": Ich gebe eurer Katze Fleisch, as already said.
genitive form of plural possessive pronoun with a group of possessors: Das Fell eurer Hunde ist sehr dicht. (your dogs' fur is very dense)