[subject] + must + [verb]
[subject] + have to / has to + [verb]
[subject] + mustn't + [verb]
[subject] + don't have to / doesn't have to + [verb]
Must and have to + verb say that something is necessary. There is no choice.
Must has one form for every person. Have to changes: I have to, she *has to*.
In the present, they are very close. Use must when you decide it is necessary, often for yourself. Use have to when the rule comes from someone else: a boss, a school, the law.
For the past and the future, only have to works: had to, will have to.
Have to makes questions and negatives with do, does and did.
These two mean very different things.
Mustn't means "don't do it, it is not allowed".
Don't have to means "it isn't necessary, but you can if you want".
In British English, have got to means the same as have to, mostly in speech: I've got to go now.