Could (Be able to/manage to)

Summary:
"Could" is used:
1. as the past form of 'can'
2. to express possibility/offer
3. to express probability, present or past
4. to express a request in the present

1. Past tense of "can"
There are at least two different ways to express the past tense of "can" for ability: could and was/were able to (meaning 'managed to'). For example:
a) He could hear well when he was younger.
b) I was able to/I managed to hand in my report in time.

In the first case (could) we are talking about general abilities, and in the second case (was able to, managed to) - about success on a particular occasion. More examples:
a) She could talk before she could walk.
b) At the conference she managed to talk non-stop for an hour without saying anything.
c) He could swim when he was very young.
d) When she went shopping yesterday she was able to walk for about 6 hours straight but now she is tired after standing for 5 minutes.

The negative couldn't can be used in both cases:
a) I couldn't hear you very well, you were too far.
b) I couldn't hand in my homework in time.
c) I couldn't sleep well last night.

2. Offers, possible actions. Used in the present:
a) What do you want to do today? We could go to the zoo or we could ride bikes in the park.
b) If you are not busy, you could help me do the dishes.
c) You could invite your friends over.
In all of the above "can" is also possible, which makes them sound more definite and certain.

*When we can see now that something was possible in the past, we use "could have done". That is, we do not change "could", but add to it the so-called perfect infinitive, which is "have" and the past participle (third form of irregular verbs):
a) Why didn't you buy the dress? You could have bought it, you had enough money.
b) You should have set an alarm, you could have overslept.
c) That was a very dangerous situation, they could have crashed.
d) I didn't know you had time last night, we could have met.
Negative:
e) He could not have come, he was not in town.

3. Probability. In this case "could" can be substituted by "might" or "may":
a) I can't find my glasses. They could be in the car again.
b) The weather is nice now but it could rain in the afternoon.
c) Pick up, it could be something important.

*Past tense in this case is also "could have done". In this way we speak about something that we assume (now) that something might have happened (before); there is a probability that it happened:
a) My glasses are nowhere to be seen. I could have left them in the car.
b) The street is wet. It could have rained.
c) Don't buy it yet, wait until after the holidays, your parents could have already got it for you as a Christmas present.
d) He could have forgotten.
Negative:
e) He couldn't have said that!

4. Request. Can be replaced by "would".
a) Could you wait a moment, please?
b) Could you hand me the salt, please?
c) Could you put me through to Mr. Jones, please?

When you are asking for something, e.g. permission (with "Could I ...?"), then "could" cannot be replaced by "would". For example:
a) Could I talk to Mr. Jones, please?
b) Could I see you for a minute?
c) Could I use your cell phone?