"Waaaait. Doesn't that mean you're just bi, then?"

A friend of mine explained to me that she was pansexual, and that it meant that she fell in love with people's personalities regardless of their gender and such.
Pretty much the same as your explanation. I don't think there's a way to shorten it

 

Your way sounds good, GreenPanRose, especially because it's what your sexuality means to you. Even if someone else had another explanation, I think you're the best person to know how to explain your orientation to people you're talking to.

If you are encountering "Oh, but that's just bi" reactions, maybe you should briefly explain why you label yourself pan and not bi -- by saying you believe there are more than two genders, for example. It can get tiresome to give people a lecture on gender and sexuality, yes, so I absolutely see your point. If it gets on your nerves too much (it does not have to be your responsibility to educate people) try suggesting that they Google it -- really, all the information they need is on the net.

 

Bi: "I can be attracted to both men and women."
Pan: "I can be attracted to anybody."

It's a vague line, and for a while I didn't really understand it, but what the others said pretty much hit all of it. It's simply a more inclusive label.

As for explaining multiple genders, it's not as complicated, I don't think. I usually just say that some people are uncomfortable with being put into categories, which is why some use "pan" (all) instead of "bi" (two), in order to allow those who feel uncomfortable with categories (non-binary people, essentially) feel acknowledged and... well, more comfortable.