Modal Verbs

Main characteristics

1) Modal verbs are followed by a bare infinitive (except for have to and ought to, which both are known as semi-modal verbs). E.g.:  
I must do my homework 
They may be in the house
but:  She ought to give up smoking

2) They don't need auxiliary verbs (except for have to). E.g.:  
Can you do it?
You cannot swim
but: She doesn't have to come tomorrow 

3) They admit contractions (except for may and have to). E.g.:
We can't speak Japanese
They mightn't work next week
but: You may not come in here

4) They never add 3rd-person singular -s ending in the Present Simple (except for have to):
Peter must work harder
She can jump very high
but: My girlfriend has to wear a school uniform

5) Some are defective: must and can. E.g.:
I can go today
I could go yesterday
I'll be able to go tomorrow
I must do it now
I had to do it yesterday
I'll have to do it tomorrow


Expressing Ability: CAN, COULD & BE ABLE TO
1) Present ability: Can
I can speak English

2) Past ability: Could
She couldn't swim when she was ten

3) Ability in other tenses: Be able to 
We'll be able to pass English next term (Future)
They have been able to escape (Present Perfect)

Expressing Obligation: MUST vs. HAVE TO
1) Internal obligation: Must (in general expresses what the speaker thinks  is necessary)
I must study for tomorrow's English exam (= I want to do it)

2) External obligation: Have to (in general suggests that somebody else has imposed the decision)
I have to be early at home  (= my parents oblige me)

However, there's no difference between must and have to in American English. In British English, Have got to is often used in informal language instead of have to (colloquial short form gotta also used in American English, though).
I have to wear a helmet at work = I must wear a helmet at work (AmE)
I have to get up early = I've got to get up early = I've gotta get up early = I gotta get up early

Expressing Lack of Obligation: DON'T HAVE TO & NEEDN'T
I don't have to wear a uniform at my school
I needn't go to school in the evenings

Expressing Prohibition: MUSTN'T
You mustn't smoke in the school
We mustn't play football inside the hotel facilities

Expressing Advice and Recommendation: OUGHT TO, SHOULD & MUST
I think you ought to follow a low-fat diet (+ formal, rare in AmE)
You should come with us and forget about your problems (- formal)
In my opinion, you must speak to her (informal)

Expressing Certainty and Possibility: MUST, MAY, MIGHT, COULD, CAN'T
1) Certainty (100% sure): Must
Someone is knocking on the door, it must be mum

2) Possibility (50% likely): May
Peter may be in his office now 

3) Probability (25% likely): Might & Could
They might come to the match with us tomorrow

4) Impossibility (0% likely): Can't
They telephone is ringing, but it can't be dad

Notice:
NEED is not a modal verb (it's lexical). It expresses necessity and can be followed by either a noun phrase or a verb (to-infinitive). Remember the lack of necessity is conveyed by means of DON'T NEED (TO) (kind of similar to don't have to when followed by a verb), and NEEDN'T is a different form (it is a modal verb) to express lack of obligation (same as don't have to):
I need some more apples 
I need to buy some more apples (it's necessary)

I don't need to buy any more apples (it's not necessary)

I needn't buy any apples (I'm not obliged to)

Why not practise here?
Exercise1, Exercise2, Exercise3, Exercise4, Exercise5, Exercise 6, Exercise7.