-you are showing off because you are excited for your friends to meet your partner and vice versa. you are excited about the relationship and want to ingratiate her with your circle of friends
-on the other hand i've definitely felt like at times guy friends were really just showing off the girl as a sort of trophy of a sexual conquest. sort of like, "look how hot this girl i'm hooking up with is"
If I'm with her, I love her, on some level, and respect her, but I know my friends, and I don't want her to meet them for a long time, until she is pretty sure we want to be a couple, because guy friends can be kind of d-bags to GFs!! Most of my friends are more 'NERDY' and might just talk her into falling asleep, standing up!!
I never show off a girl unless we are in a relationship that I'm very excited or serious about. It's not really my thing to show off girls who aren't going to stick around because their hot. I don't like how it makes me look. I like to give off the impression that all the girls I date are awesome so normally if I'm showing one off, it's because I'm really proud of what a cool person they are AND that I think their attractive.
I'm proud of my girlfriend if I show off to my friends. Simple as that. Doesn't mean I'm NOT proud if I DON'T show off, because it depends if my friends are going to treat her like a trophy I won or like a human being.