These words are all used for obligations - things that are necessary and required.
Must is the most formal. It is usually used in official rules (and is not as common in spoken English):
Students must register for classes by August 1.
You must possess a valid driver's license to apply for this job.
NEVER use the word "to" after must. "Must to" does not exist!
In everyday English, need to, have to, and got to (informal) are more common ways of expressing obligations:
We need to buy some milk - there's none in the refrigerator.
My son needs to get a haircut; his hair is getting really long.
I have to leave work early today in order to pick up my kids from school.
Sarah works at a restaurant, so she has to work nights and weekends.
We've been really stressed out lately; we gotta (got to) take a vacation.
When talking about actions you need to do, ALWAYS use the word "to" after need/have and before the verb:
We need buy some milk
I have leave work early
When spoken fast:
"need to" sounds like needa
"needs to" sounds like needsta
"have to" sounds like hafta
"has to" sounds like hasta
"got to" sounds like gotta
Be careful when using the negative form. Mustn't is different from "don't need to" and "don't have to." Mustn't means something is prohibited/forbidden, but "don't need to" and "don't have to" mean something is not required; it is optional.
You mustn't smoke inside the hospital.
You don't need to buy a ticket; the concert is free.
You don't have to go to the extra review session; it's an optional class.
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