Modals

Modals are verb, special verbs, we use a lot. They behave differently because they show if we believe something true, probable or possible, or not at all. We also use them when we are doing thing like talking about ability, asking permission, making request or offers, etc...

Modals Verb:

Can (Not): "I can speak English".
Could (Not): "I could speak English".
May (Not): "I may speak English".
Might (Not): "I might speak English".
Must (Not): "I must speak English".
Need to: "I need to speak English".
Ought to: "I ought to speak English".
Shall (Not): "I shall speak English".
Should (Not): "I should speak English".
Will (Not): "I will speak English".
Would (Not): "I would speak English".

We need to take into account that:
We do not use and "s" when we are talking in third person singular.
"She can go to the party".
We use them to make questions so we need to do an inversion.
"Can I go?"
They are always followed by another verb which must be in infinitive without the "to".
"I must go"

There a lot of situations where we use them:

Probability: We can use them when we want to say how sure we are about something happening. That is why some people call them "modals of deduction".
"It´s snowing, so it must be very cold outside".
Ability: When we are talking about a skill or an ability.
"I can´t drive my father´s car".
Obligation and Advice: We use "must" or "should" when we mean something can be necessary or unnecessary or to give advice.
You should stop smoking.
Permission: We use verbs as "can", "could" and "may" when we are asking for a permission or when we say that something is not allow to do.
"Could I drive?"
"You may no use it today".
Habits: We use "will" and "would" when we are talking about habits or things we are used to do or did.
"When I was younger, I would often eat a lot of ice cream".
"Mary will always sleep late".

Uses of WOULD

Would is an auxiliary verb, we use it when we are talking about the past, future in the past and for express the conditional mood. We also use it to express desire, polite request and questions, opinions, hope, wish and regret.

Uses:

Talking about the past: "Even as a boy, he knew he would succeed in life".
Future in Past: "She met a man that she would one day marry".
Conditionals: "If he lost his job he would have no money".
Desire: "I´d love a glass of water".
Polite: "Would you open the door for me, please?".
Opinion or Hope: "I would imagine that they will by a new house".
Wish: "I wish I he would be here".
Presumption or Expectation: "That would be Jo calling, I´ll pick up".
Uncertainty: "He would seem to be getting better".
Derogatory: "They would say I came, wouldn´t they?"
Regret: "Would that it were true!" (This is rare, poetic or literary, it is not that formal).