Expressing ability

Can and be able to
Can and be able to are both used to express ability. However, can only has present tense (can) and past tense (could) forms. If another form of the verb is required, be able to is used.
Present: She can/is able to speak two languages.
Past: She could/was able to read when she was three.
Infinitive: I'd like to be able to run.
Present perfect: She's never been able to save money.
Will future: He'll be able to drive when he passes his test.

Present ability
We use can or be able to to talk about present abiity. Be able to is more formal than can.
The negative form of can is can't or cannot. To form the negative of be able to, not is used before able. You can also use be unable to.
Many of my relatives aren't able/are unable to pronounce the word "taught" correctly.
Be capable of + gerund can also be used to express ability. It means to have the ability, capacity or potencial to do something.
The team has some excellent players and is capable of winning the championship.
The negative form be incapable of + gerund can be used or not can be placed before capable.
He is incapable/isn't capable of looking after himself.

Past ability
When we talk about general ability in the past, both could and was/were able to are possible.
He could paint really well when he was a child.
Bill wasn't capable of making toast without burning it.
When we talk about ability to do something on one occasion in the past, could is not possible. Instead, was/were able to, managed to + infinitive or succeeded in + gerund have to be used.
We managed to/were able to speak to Bill last night.
However, could can be used for ability on one occasion when it is used with verbs of the senses: see, smell, feel, hear, taste, sense.
I knew my husband has arived; I could smell his perfume.
When we talk about inability to do something on one occasion in the past, couldn't, weren't/wasn't able to, didn't manage to and didn't succeed in are all possible.
I couldn't do the homework last night; it was too difficult.

Permission

To express permission it is possible to use can, may (more formal) or be allowed to. In the negative these express lack of permission, or prohibition.
You can order another drink but you can't have any more hamburgers. 
We aren't allowed to wear trainers to school.

May is not possible in the past. Could and be allowed to are possible for general permission.
In my last job we had flexitime so we could arrive more or less when we wanted to.

Could is not used when referring to a particular situation in the past. Only be allowed to is possible.
I was allowed to stay up late last night.

Can, could, be able to (ability and possibility) II

Más información y ejercicios sobre estos modal verbs:

http://www.englishclub.com/grammar/verbs-modals_can.htm

http://elblogdelingles.blogspot.com/2006/05/lesson-17-can-could-and-be-able-to.html

http://www.autoenglish.org/gr.can.i.htm

Can, could, be able to (ability and possibility)

Can / Could
I can speak Spanish very well.
She could play the violin when she was three.
Could you open that door, please?
Can is a modal verb. It only has a present, past, and conditional form (but can also be used with a future meaning).
For other tenses and forms use be able to.

Be able to + Infinitive
I am able to accept your invitation.
They weren't able to come.
I'll be able to practise my English in London.
I'd like to be able to ski.
You can use be able to in the present, past, future, present perfect, and as a gerund or infinitive.
Be able to in the present and past is more formal than can/could.