Being "hung up on" something means to be really worried or upset about something, so that you can't stop thinking about it. You can "be hung up on" a person, which usually means that you have romantic feelings for that person that aren't being returned. Or you can get "hung up on" an idea:
Don't get hung up on the details. Just try to do your best.
People get so hung up on brands and labels.
There's a totally different meaning of "hung up on" as well. When you "hang up on" a person, it means that you end the phone conversation with them suddenly:
He hung up on me! What an asshole!
Notice that that's different from:
He's hung up on me.
...which means that he's obsessed with you or in love with you.