A nick-name for a person who always tries to appear better then their peers. Most commonly noticed in the form of story telling past experiences, which are most often lies.
The "green" aspect of it is a reference to the green mushrooms of any super mario game that gives you "1-up". By definition, a greenie will always try to "1-up" whatever you've just said/done to conpensate for their own lack of self worth.
Angel:I just got a new job, pays really well and I think I found a relly nice place to live with some good roomies.
Zak: yeah, I solved world hunger.
Felicia: No you didn't. Be happy for her and stop being such a 1-Up'ing greenie s.o.b.!