One of my eyes is bigger, or differently shaped than the other, and it is quite obvious. People have brought it up in pictures (it especially shows in pictures), or when I'm sitting across the table from someone and make eye contact.

I have astigmatism. I also used to wear contacts 10 hours a day (10 years ago - though now, I only wear contacts for about 5 hours each week TOTAL).

But can someone tell me, what causes different eye shapes? Will Laser surgery correct the problem? Any other solutions? (Other than makeup - I mean, are there any PERMANENT solutions to fix this?)

It's genetic (I'm assuming you don't mean one eye has suddenly changed shape). No one's face is exactly symmetrical. Will laser eye surgery fix the problem? No, because your problem is anatomical, not visual. Perhaps you could consult a plastic surgeon. I'm not sure there is much they could do, but it wouldn't hurt to find out, if this really bothers you. But I suspect it is something you will just have to accept. It's not that uncommon, either. Ever see pictures of model Kristy Turlington? She has noticebly different-sized eyes and it didn't hurt her career as a supermodel!