First of all, it's deasupra (not deasupră). Like many adverbs of this type, there is no căciulă over the last "a." The difference between deasupra and pe deasupra is the same as the difference between sub and dedesubt. With deasupra, you say, "deasupra + a noun" (e.g. on top of the car, on top of the table, etc) but "pe deasupra" is just "on top" (but not followed by a noun). So, I put the book on top (not on top of the table).
Pe deasupra can also be used to mean "besides," depending on context.
It's just the rule for personal pronouns following prepositions taking the genitive. So mea, ta, noastră, voastră for 1st and 2nd person, but lui, ei, lor for 3rd person.
That's if it's following a feminine form, eg "deasupră". If it's following a masculine form, eg "în jurul", then you'd use meu, tău, nostru, vostru. Or lui, ei, lor again.