Translation:I am the leader of this party, you are just a member.
Roughly, you would use genitive+possessive when English would use "X of the Y" or "the X's Y", and nominative+possessive when English would use "the X Y".
Here it's "the leader of this party" / "the party's leader", so the party is in the genitive.
If it had simply been "the party leader" (not a specific party's leader, just a party leader in general), then nominative+possessive would have been appropriate.
Compare, for example, şehir merkezi "the city centre" and şehrin merkezi "the city's centre" (i.e. the centre of a particular city).
As for partinin lideriyim, you have partinin lideri "the party's leader" and then -yim for "I am ..." to get partinin lideriyim "I am the party's leader".