can or could

Possibility

We use the modal can to make general statements about what is possible:

       It can be very cold in winter. (= It is sometimes very cold in winter)
       You can easily lose your way in the dark. (= People often lose their way in the dark) 

We use could as the past tense of can:

       It could be very cold in winter. (=Sometimes it was very cold in winter.)
       You could lose your way in the dark. (=People often lost their way in the dark)

We use could to show that something is possible in the future, but not certain:

If we don’t hurry we could be late. (=Perhaps/Maybe we will be late)

We use could have to show that something is/was possible now or at some time in the past:

       It’s ten o’clock. They could have arrived now.
       They could have arrived hours ago.

Impossibility:

We use the negative can’t or cannot to show that something is impossible:

       That can’t be true.
       You cannot be serious.

We use couldn’t/could not to talk about the past:

       We knew it could not be true.
       He was obviously joking. He could not be serious.

Ability:

We use can to talk about someone’s skill or general abilities:

       She can speak several languages.
       He can swim like a fish.
       They can’t dance very well.

We use can to talk about the ability to do something at a given time in the present or future:

       You can make a lot of money if you are lucky.
       Help. I can’t breathe.
       They can run but they can’t hide.

We use could to talk about past time:

       She could speak several languages.
       They couldn’t dance very well.

Permission:

We use can to ask for permission to do something:

       Can I ask a question, please?
       Can we go home now?

could is more formal and polite than can:

       Could I ask a question please?
       Could we go home now?

We use can to give permission:

       You can go home now if you like.
       You can borrow my pen if you like.

We use can to say that someone has permission to do something:

       We can go out whenever we want.
       Students can travel free.

Instructions and requests:

We use could you and as a polite way of telling or asking someone to do something:

       Could you take a message please?
       Could I have my bill please?

can is less polite:

       Can you take a message please?

Offers and invitations:

We use can I … to make offers:

       Can I help you?
       Can I do that for you?

We sometimes say I can ... or I could ... to make an offer:

       I can do that for you if you like.
       I can give you a lift to the station.