[chryuop]

Princess, don't look at the の as a "of" translation because it doesn't work with that translation 100% of the times. There are typical phrases that you find on mostly every text book that will give you a bad migraine if you keep translating as "of": 先生の人、男の人、女の子 and many more.
I got used to seeing that construction more like a subordinate phrase, that helps me more. I don't know what level is you Japanese knowledge at, but if you have already studied subordinate clauses you will understand what I mean.
The examples above could be read as "the person who is a teacher", "the person who is a man" or "the kid who is a female". The only difference is that for the verb "to be" you cannot use it normally after a noun like for other verbs (like for example りんごを食べた人 the person who ate an apple). For the verb to be you can have either である or の.
I hope I won't be misleading.