What does it mean to be apart of a community? what does it mean to be apart of a family?

First of all, I think you mean "a part". To be "a part of a community" means that you are a member of a social group or circle. To be "a part of a family" means you are a member of a family. However, when you do not include the space between "a" and "part" and simply use the word "apart" the meaning is changed and the sentence is grammatically incorrect. The word "apart" implies that you ARE NOT a part of whatever you are talking about. However, if you want to say that you are "APART" and that is the word you mean, you would say "APART FROM a community" or "APART FROM a family" meaning you are different than the community/family.
I hope this makes sense!
Good luck!