Both can and could are modal auxiliary verbs. Can is used to talk about present or general ability. Could is used to talk about ability that existed in the past.

Compare:
She can knit.
He can speak ten languages.
She could write when she was three.

Note that we use will be able to and not can to talk about future ability.

You will be able to speak good English in a few months.
One day scientists will be able to find a cure for cancer.

In reported speech

Could is the past tense form of can in reported speech.

‘Can you help me,’ she said to him. (Direct speech)
She asked him if he could help her. (Indirect speech)

Possibility
Can is used to talk about theoretical possibility.

Glass can be blown.
Accidents can happen any time.

We use may, might or could to talk about the chances that something will actually happen.

It could rain this evening.
We may/might go camping this summer.

Can is used in questions and negatives to talk about the logical possibility that something is true.

‘There is the doorbell.’ ‘Who can it be?’ ‘Well, it can’t be the postman.’

Note that can is not normally used in affirmative sentences with this meaning. Instead we use may, might or could.

‘Where is John?’ ‘He could/may/might be at the library.’ (NOT He can be at the library.)

Permission

Can is used to ask for and give permission. Can’t is used to refuse permission.

‘Can I have a look at your papers?’ ‘Yes, of course, you can.’
‘Can I have another cup of coffee?’ ‘No, I am afraid, you can’t.’

Could is also used to ask for permission. Note that could is more polite than can.

‘Could I have a look at your papers?’