My fish, Gyarados chose me (pictured). When I went to the pet store, I didn't care for his color (which was mostly red with a TINY bit of blue that was barely noticeable). There were better looking fish like halfmoons and crowntails. But he was the most active fish and he kept following me in fast, swift motions and would snap his mouth at me whenever I got close. I found myself continuously looking at him, despite me being unimpressed with his colors (and look at them now!) and I told myself that if I couldn't help but look at him one more time, I'd get him. And it happened. XD
But yes, you want to have active fish (not ones that look like they're hurting themselves like flipping out, banging against the sides repeatedly), vivid and bright colors, no flat fins held close to the body, spread out, and aren't floating at the top or laying on the bottom even when you approach.