There are two kinds of verbs in English: special verbs & normal verbs. There are 13 special verbs, all the rest are normal verbs.

NORMAL VERBS
They can only make affirmative sentences. For all the other things, they need DO:
affirmative I live in London
negative I don't live in London
interrogative Do you live in London? Yes, I do.

SPECIAL VERBS
They never use DO, they can do everything. Most of them can use contractions.
affirmative I can speak English
negative I can't speak English
interrogative Can you speak English? Yes, I can.

The special verbs are: to be, have, do, can/could, will/would, shall/should, may/might, must, ought to

There are two kinds of special verbs: auxiliaries and modals.

AUXILIARY VERBS (or HELPING VERBS)

The auxiliaries verbs in English are: to be, have, do, will, would.

They help a normal verb to make some tenses. A normal verb only has present and past tense, but we can make more tenses using auxiliary verbs:

present: I live in London
past: I lived in London
present perfect : I have lived in London auxiliary: Have.
future: I will live in London auxiliary: Will.
conditional: I would live in London auxiliary: Would
present continuous: I am living in London auxiliary: Be.
passive voice: I am called Tim auxiliary: Be.
present perfect continuous: I have been living in London auxiliaries: Have & Be.
negative: I don't live in London auxiliary: Do.
etc.

To Be to make continuous forms and the passive voice: I am writing a letter  -  The window was broken
Have to make the present perfect: He has lived here for 5 years
Do to make negatives, interrogatives, etc: I don't like tennis / do you know him? Yes, I do.
Will to express future: They will come to visit me
Would to make conditionals: If you had problems, I would help you.

MODAL VERBS

Modal verbs are used to express ideas such as possibility, intention, deduction, suggestion, obligation, etc.
Modal verbs never change, they don't add an -S for the 3rd person singular and they don't use TO before or after them (exceptions: to be, ought to):
normal: I live here, he lives here
modal: I can swim, he can swim
normal: I want to go
modal: I may go

POSSIBILITY: May & Might
- It may rain today because it is very cloudy.
- If you go to Hollywood you might see some famous actors in the street.

PERMISSION: Can, May
- Can I go to the toilet?
- May I go to the toilet? (more polite)

ABILITY: Can (past tense and conditional tense: Could)
- I can play the guitar
- When I was a child I could play the guitar, but now I can't.

ADVICE: Should
- If you want to be a doctor you should study more.

INTENTION: Will (past: Would)  [yes, this verb can be a modal or an auxiliary for the future)
- Don't worry, I will help you.
- He said that he would help me.

INVITATION: Will
- Will you come to my party tonight?
- Sure, I will.

DEDUCTION:  Must ("can't" for the negative)
- Who's that? - That must be John, because he is very tall.
- No, that can't be John, John is younger.

SUGGESTION: Shall
- Shall we go to the cinema?

OBLIGATION: Must
- You must stay here and you mustn't speak to anyone until I come back home

PROHIBITION: Can (in the negative)
- I'm sorry, you can't park here, there is a yellow line on the road.

SUMMARY

The 3 No's: Remember this simple rule for the use of special verbs (modals & auxiliaries): NO -S, NO TO, NO DO

NO –S I can/ she can
NO TO I can walk / no infinitive
NO DO you can / you can’t / can you?