'Have to' means that it is necessary for you to do this.
'Supposed to' means that you are expected to do this.
For example, you might say this:
'I'm supposed to stay at work until 5 pm, but I have to go home now. I feel really ill.'
You are 'supposed to' stay until 5 pm because this is what is expected of you, what you have been told to do, or what it says on your contract. This form is often used as a contrast to a different fact, about what you want or have to do. In this case, even though you are supposed to work until 5 pm, you have to/need go home now because you feel too ill to work.
'Must' has a similar meaning to 'have to', but it is less common. It's often used as a stronger form of 'I think you/I/we should...'.
In fact, in spoken English, the commonest use of 'must' is to indicate certainty, as in 'You must be exhausted'.