Em inglês, verbos auxiliares modais são verbos que só ocorrem na presença de outro verbo, são defectivos na conjugação e não têm passado nem futuro (com exceção do can que tem passado e condicional).
Devido à alta freqüência com que ocorrem na língua, os verbos modais tornam-se imprescindíveis.

Veja aqui uma lista dos principais:

Can - Could

- significado de ability: I can speak English - Eu consigo falar inglês.
- significado de permission: Can I smoke here? - Posso fumar aqui?
- significado de possibility: It can happen to anyone. - Isso pode acontecer com qualquer um. (Esta ocorrência é mais rara e o significado de possibilidade aqui se confunde com o de capacidade. Para possibilidade é sempre melhor usar may e might.)
Veja aqui um rápido estudo sobre a diferença de pronúncia entre can e can't.

Could funciona como passado e como futuro do pretérito de can:

- passado: I couldn't speak English before Going To England. / I co uldn't go. / You couldn't smoke in the presence of your parents at that time. - Eu não sabia falar inglês antes de ir para a Inglaterra. / Não pude ir. / Não se podia fumar na presença dos pais, naquela época.

- futuro do pretérito: You could have called me. Could you do me a favor? - Você poderia ter me ligado. / Você poderia me fazer um favor?

May

- significado de permission: May smoke here? - Posso fumar aqui?
- significado de possibility: It may rain today. - Pode ser que chova hoje.
- para expressar um desejo (to express a wish): May all your dreams come true. - Que todos seus sonhos se realizem. (Esta última ocorrência é mais rara, restrita a uma linguagem mais formal.)

Might

- significado de remote possibility: It might rain this weekend. - É capaz de chover no próximo fim de semana.

Should

- significado de advice: You should study more. - Você deveria estudar mais.*

Shall

- significado de suggestion (predominante no dialeto britânico): Shall we go to the movies? - Que tal irmos ao cinema? / Que tal, vamos ao cinema?

Shall ocorre unicamente no modo interrogativo e na primeira pessoa do singular (I) ou do plural (we).

Must

- significado de obligation: You must stop smoking. - Você tem que parar de fumar.
- significado de prohibition: You mustn't get out of bed. - Você não pode sair da cama.
- significado de inference, logical deduction: He must be very rich. - Ele deve ser muito rico.*
* Veja a ambivalência do verbo dever do português.

Há quem classifique o will, o would e o used to como verbos modais. Nós preferimos deixar o will e o would como auxiliares do future e do conditional, e o used to como habitual, equivalente ao pretérito imperfeito do português. Leia aqui sobre o used to.

Também o verbo need pode ocorrer como modal, mas apenas nas formas negativa e interrogativa. Esta, entretanto, é uma ocorrência muito rara, principalmente em inglês norte-americano.

VERBOS MODAIS

Ausência de Obrigação

Quando nós queremos dizer que não é necessário a pessoa fazer determinada ação nós podemos utilizar um verbo moldal no negativo  o verbo inteiramente no negativo. Mas nós podemos utilizar o don't have ao invés do haven't got to, que é mais informal.

It is said that Sunday is the best day of the week because you don't have to work and you haven't got to wake up early. You don't need to have breakfast quickly and you can read the newspaper as long as you want. This is true, but today is Sunday and I am in my utility room. What can I see? An iron, a broom, a washing machine, a full laundry basket, a feather duster and a vacuum cleaner.
 
What do I use them for? To do the housework. This means that I will have to spend the whole Sunday using the window cleaner, the mop and the dustpan. But people say that on Sunday you don't have to work and you haven't got to wake up early. You don't need to have breakfast quickly and you can read the newspaper as long as you want. So, what is happening? Maybe people forget that some Sundays you must work at home. It is true you don't have to go to work to your office or to your shop but the laundry detergent, the bleach and the softener are waiting for Sunday to be used.

Ability

Modals are auxiliary verbs which are used with a bare infinitive to express the speaker's attitude as well as a degree of certainty or uncertainty. Each modal verb has more than one use. When we want to say that someone has the ability to do something we usecan, but ascanonly has two forms —canfor the present andcouldfor the past —be able tosupplies the missing parts ofcanand it is also an alternative form for present and past.

An afternoon at home

The Robinsons are spending the afternoon at home.

Father– Hey, Peter. I think your brother has some problems with his maths exercises. Why don't you stop playing with the logs and the fireplace and go to his room to help him?
Peter– Can't he solve his maths problems himself?
Father– No, he can't. I'd like to help him, but you know I can't, because I've forgotten what I was taught when I was at school. And your mother could help you when maths was easier. Now it is very complicated.
Peter– Mum, what are you doing?
Mother– I'm sewing the curtains.
Peter– When are you Going To teach me how to sew?
Mother – Very soon. If you help your brother, I'll teach you to sew. There are not many boys of your age who can sew. And don't talk so loud. The baby can't sleep, because you are making too much noise.
Peter– Dad, what are you doing?
Father– I'm very busy. Yesterday mum told me something about changing the armchair and the sofa, and I'm just trying to convince her that they are still comfortable. You know it's difficult to convince your mother. Please, go and help your brother. You are the only one who can do it. Your mother and I don't remember anything of what he is doing.
Peter– OK. I'll do it. Mum, don't forget to tell daddy that the TV and the video cassette recorder don't work properly and that it would be a good idea to change them too.