I feel bad about someone = this is not really used often. I would say it means I have a bad feeling towards this person or I feel some guilt about something I did to them. But I wouldn’t use this expression. It would be more common to say “I feel bad for someone” which means I feel sorry for them, I feel pity towards them. I feel bad about something = I did something bad and I feel guilt or remorse about it. Or I feel bad about a bad thing that is going on in the world. I feel bad for you= as I wrote above, I feel sorry for you, I feel pity towards you, I have sympathy for you.