The most common modal verbs we use to talk about things that are mandatory are must and have to.

In the present and in the affirmative, both have very similar meanings: obligation.

However, must implies that the obligation comes from within; i.e., you are obliging yourself to do it.

I must get to the professor’s office before he leaves. (This is what I feel, or the reasons for the obligation are personal; e.g., I have a question I need to ask him that I couldn’t ask him in class)
I must remember to stop by the gas station tonight. (No one but me is making me—I don’t want to run out of gas!)

On the other hand, have to implies that the obligation is external; i.e., someone else is obliging you.

I have to get to the professor’s office before he leaves. (Someone else is making me do this; e.g., the professor told me to turn something in before the end of the day).
I have to remember to stop by the gas station tonight. (Someone else asked me to fill up the car).

Oftentimes, which you choose to use depends entirely on how you see the situation.

In the negative, however, these two verbs are entirely different:

Don’t have to means that something isn’t necessary:

You don’t have to finish your homework tonight if you don’t want to.
Peter doesn’t have to study tonight because he passed his exam.

Mustn’t means that something is prohibited:

You mustn’t forget to do your homework tonight!
Peter mustn’t stop studying tonight because he failed his exam.