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"Would" is used to declare a preference and to ask for something politely.
EXAMPLE
I would like a beer and my wife would like a glass of wine please.
I would like to play tennis on Sundays.
I would never go to that city.

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"Should" indicates an obligation or recommendation. Reflects an opinion about what is right. It is translated as the conditional of "duty" in Spanish.
EXAMPLE
I should  see a doctor. I have a terrible headache.
You should revise your lessons.

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"Ought to" is a synonym of "should".
EXAMPLE
He should / ought to be very tired. He's been working all day long.
They ought to work less.

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"Shall" is used as "will" to form the future tense. The use of "shall" is much more common in the United Kingdom and is generally more educated.
EXAMPLE
Chris shall be happy to see you.

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We use this modal to say a strong obligation.
Also, we use this modal for a logical conclusion / Certainty.
EXAMPLE.
You must stop when the traffic lights turn red.
He must be very tired. He's been working all day long.
I must go now.

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We can use this modal to express possibility in the future.
Also we can use it to request permission.
EXAMPLE
My I go to the bathroom please?
I my be home late.
You may leave if you like.

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We use this modal to indicate possibility in the present or future.
In this cases, "Might" is synonymous of "May".
EXAMPLES
Might I suggest an idea?
I might go on holiday to Australia next year.

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"Will" is used to form the future tense. Also the use of "will" means will or determination.
EXAMPLE
I will help you.
I will be late to my house.

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Could is the past of can, indicates ability or ability in the past.
- ability in the past
- polite permission
- possibility
EXAMPLE
When I was younger I could run fast.
Excuse me, could I just say something?
It could rain tomorrow!

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We use this modal to express ability.
Also we can use this modal to indicate a possibility  or request permission.
EXAMPLE
I can swim.
Can I use your phone please?
Smoking can cause cancer

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Have to is NOT an auxiliary verb (it uses the verb have as a main verb). We include have to here for convenience.

Must is a modal auxiliary verb.
Have to expresses impersonal obligation. The subject of have tois obliged or forced to act by a separate, external power (for example, the Law or school rules). Have to is objective.
EXAMPLES
In France, you have to drive on the right.
In England, most schoolchildren have to wear a uniform.
John has to wear a tie at work.