Could

Present:

Possibility

We often use "could" to express possibility in the present or future. This is distinct to expressing certainty or fact. Compare:

It is Colin, only Colin has that bark (I am certain that is my dog, I recognise his bark)

It could be Colin, it sounds like him (There is a strong possibility it is my dog, it sounds like his bark, but I cannot be certain)

See also: can

Suggestions

We are able to use "Could" to make or express suggestions. It is most common to do this in a question form:

You know the party is fancy dress don't you?

Oh, I wasn't sure

What will you go as?

I suppose I could go as Doctor Who

What? Again? You could probably try and come as a nother character this year.

Well, it makes no difference anyway, there is no way I am going to be able to get back in time.

Well, could you catch an earlier train?

I could do, I will need to speak to my boss.

Permission

Could is used to ask for permission. It is a more formal or polite form to ask than using can.

Could I leave work early on Friday?

Could I ask you a personal question?

We DO NOT use could to give or refuse permission, that is with can/may

Could I leave work early on Friday?

You may not. I need someone needs to meet the Americans at the office.

See can/may

Past

Ability

We use "could" to talk about general past ability:

I could draw really well when I was at art college

I couldn't score penalties when I was younger

To talk about ability or lack of ability on a specific occassion we use "was/were able to" rather than "could":

He was able to score the penalty in the last minute of the game

NOT

He could score the penalty in the last minute of the game

Single events or actions

While we use "able to" or "managed to" not "could" to talk about ability on a specific occassion, we can use "could" to refer to specific abilities or actions related to sense or states.

We went to that new restaurant and the food was terrible. All I could taste was garlic

I knew we had to get out of the house when I could smell smoke.

She came to speak to me but I simply could not understand what she was saying

We could sense he was going to say something important when he asked the room to be silent. 

See also be able to

We don’t usually use could to talk about single events that happened in the past because these are facts rather than possibilites, but we can use "could not" in negative structures:

I was able to/managed to get a job after graduating.

NOT: 

I could get a job after graduating

But:

I wasn't able to/didn't manage to get a job after graduating.

AND

I couldn't get a job after graduating.

Possibility

We can use could have + -ed form to talk about possibility in the past:

I could have been a football player but I didn't train hard enough

I could have worked in a bank if I had studied hard enough

We could have moved to the village but we chose to stay in the city.

I couldn't have tried any harder to convince her to stay.

Guessing, Prediction (deduction)

When we want to guess or predict something, we use couldn’t as the negative form of must. We use couldn’t have + -ed form as the negative form of must have + -ed. 

That couldn't have been his wife, you must be mistaken

A dog couldn't have made all this mess, it must have been 

Couldn’t and couldn’t have + -ed form express strong possibility:

Critisicm

We often use could have + -ed form to express disapproval or criticism:

You could have told us you were not coming

You could have done your homework before you went out

Regret

We use could have + -ed form to talk about things that did not happen and sometimes to expresses regret:

I could have have been a lawyer if I had studied hard enough

This can often look and sound like expressing possibility in the past and there is plenty of cross over between the ideas.  

Reporting "Can"

In reported speech we use could when reporting can:

She told me we could reserve a room online (Reporting the phrase "You can reserve a room online")