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We use the modal 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 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.

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Meaning: synonym of “may”, indicates possibilities in the present or future

Examples:
I would bring an umbrella, it might rain later.
It might be better to finish this now, rather than wait until tomorrow.

“Might” can also be used like “may” to ask permission, but this is much more common in British English than in American English.

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We use will:

to talk about the future – to say what we believe will happen

to talk about what people want to do or are willing to do

to make promises and offers

We'll be late.
We will have to take the train.

We use will in conditionals with if and unless to say what we think will happen in the future or present:

I’ll give her a call if I can find her number.
You won’t get in unless you have a ticket.

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Used, like “will”, to express the future
The use of “shall” is much more common in British English and is generally considered more polite.

Examples:
Chris shall be happy to see you.
I’ll take the 3 o’clock train.

We also use “shall” for offers or to make suggestions or to ask about opinions or preferences.
Examples:

Shall we meet at 10pm? Shall we go to the movies or a museum?

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Used to indicate possibilities in the future

Examples:
I would bring an umbrella, it may rain later.
It may be better to finish this now, rather than wait until tomorrow.

“May” is also used to give permission or instructions.
Examples:
You may leave if you like.
You may use your cell phones now.

In interrogative sentences, “may” is the most polite way to ask permission (as opposed to “can” or “could”).
Examples:
May I have a glass of water?
May I leave now?

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Indicates a recommendation or obligation or reflects an opinion about what is right or correct
Examples:
I should call my parents more often.
You shouldn’t work so hard.
They should practice more if they want to win the championship.

“Should” is used in interrogative sentences to ask if there is an obligation or to ask for a recommendation.
Examples:
Should we leave a tip? Should I have the steak or the chicken?
Where should they meet you?

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We use the modal must to show we are sure something to be true and we have reasons for our belief:

It’s getting dark.
It must be quite late.
You must be hungry.

We use must have for the past:
They hadn’t eaten all day. They must have been hungry.
You look happy. You must have heard the good news.

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would is the past tense form of will. Because it is a past tense it is used:

to talk about the past.

to talk about hypotheses – things that are imagined rather than true.

for politeness.

to say what we believed would happen:

I thought I would be late …… so I would have to take the train.

We use would to talk about hypotheses, about something which is possible but not real:

to talk about the result or effect of a possible situation:

It would be very expensive to stay in a hotel.

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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 the negative can’t or cannot to show that something is impossible:
That can’t be true.
You cannot be serious.