The verb, to hang, has to do with suspending something, such as laundry from a clothesline. You can say 'I hung up the laundry.' You can say 'I hung up the phone' meaning you ended the call. To be hung up on something or someone means that you can't stop thinking about it or them. You could say 'He is really hung up on his girlfriend.' This was more popular years ago and you don't hear it as often now. You can also say 'I got hung up at the party' meaning you were delayed, meaning you spent more time there then you intended to.
Hang up is another issue. Again, I can hang up the laundry or hang up the phone. The slang word, hangup, is a noun that was more popular a few years ago, means to have personal issues that they can't get over. 'You have too many hangups'. It was usually said when someone wanted you to do something that went against your values. When you said no because of your personal value system, the other person would tell you that you had too many hangups as a way to undermine your values and make you 'second guess' your integrity. It makes you feel like there is something wrong with you for saying no. An example would be one person pressuring someone else to drink alcohol. The person saying no would be told that they have a hangup because they don't want to drink.
Hope that helps.