Contractions are forms like I’ve and he’ll. There are two kinds of contractions:

Noun/pronoun etc + auxiliary verb

He’s coming. (= He is coming.)
I’m tired. (= I am tired.)
I’ve no idea what he is going to do. (= I have no idea what he is going to do.)
My father’s not very well. (= My father is not very well.)
There’s a problem. (= There is a problem.)

Auxiliary verb + not

They aren’t ready. (= They are not ready.)
He won’t come. (= He will not come.)
I haven’t seen him in years. (= I have not seen him in years.)

Contractions are formed with auxiliary verbs. The short form ’s (= is/has) can be written after nouns, questions words, here and now.

Here’s our bus.
My mother’s coming with us.
Where’s John?

The short forms ’ll, ’d and ’re are common after pronouns and unstressed there. In other cases we usually write the full forms.

Note that contractions are not usually written with double subjects.

Alice and I have decided to start a business.  (NOT Alice and I’ve decided to start a business.)

Contractions are common and correct in informal writing. We do not normally use them in formal writing.

List of some common contractions

I’m = I am
I’ve = I have
I’ll = I will
I’d = I would/had

You’re = You are
You’ve = You have
You’ll = You will
You’d = You would/had

He’s = He is/has
He’ll = He will
He’d = He would/had

She’s = She is/has
She’ll = She will
She’d = She would/had

It’s = It is/has
They’re = They are
There’ve = There have

We’ve = We have

We’re = We are

Aren’t = Are not
Isn’t = Is not
Doesn’t = Does not
Didn’t = Did not
Won’t = Will not
Can’t = Cannot
Hasn’t = Has not
Haven’t = Have not
Needn’t = Need not