Learn Hardcore Latin: Hilaritas aviae puerum flentem consolatur. - Grandmother’s cheerfulness comforts the crying boy.

You can tell because hilaritas is in the nominative singular, the case normally used for the subject of a finite verb. The verb consolatur is also third person singular, so it matches hilaritas: cheerfulness comforts.

So the basic skeleton is:

hilaritas = the one doing the action
consolatur = comforts

What case is aviae, and what does it mean here?

Here aviae is most naturally genitive singular, meaning of the grandmother or the grandmother’s.

So hilaritas aviae means the grandmother’s cheerfulness.

Formally, aviae could also be dative singular, since the ending -ae can represent more than one case. But in this sentence, the genitive makes much better sense:

hilaritas aviae = the grandmother’s cheerfulness
not likely cheerfulness to/for the grandmother

This is a very common thing in Latin: one ending can be ambiguous, and you decide from grammar and sense.

How do we know aviae goes with hilaritas?