To learn the basic use of the Saxon Genitive click here: Saxon Genitive.

We use 'S to express possession when talking about people:

Phoebe has a cat This is Phoebe's cat

SPECIAL CASES

We can also use it with time words and expressions:

Today's newspaper
Come to next Saturday's party
It was a three-weeks' journey

We often use it when talking about things referring to people (countries, organizations, firms...)

Italy's first president was De Nicola
Manchester United's players are in Egypt
Bill Gates revealed Microsoft's plans for the future

We can also use the 'S with animals, especially your pets and big animals

My dog's leg is broken = The leg of that dog is broken
The elephant's baby is very cute = The baby of the elephant...

  We can use the  'S  as a pronoun when the noun has been mentioned or it is understood

Is that Peter's car? - No, that's Kevin's  (= Kevin's car)
This is my book and this is Mike's

For the same reason, when the context is clear we don't mention the "thing" possessed when we are talking about some places: people's house, shop, church or firm

Yesterday I went to Susan's   (Susan's house)
We had a beer at Paco's   (Paco's bar)
They got married at St Joseph's   (=St Joseph's church)
I went to the baker's    (=the baker's shop)
I must go to the doctor's   (=the doctor's office)

In many of these cases, it's now very common to see the firm, shop or work place used without 's: I went to the doctor, We got married at St Joseph, I always buy at Sainsbury.

The possessive form of WHO is WHOSE (not who's):

- Whose car is this?  - It's mine.
- Whose telephone number is this?  - It's Mike's

Don't be surprised if you find the construction OF used with people even in short sentences:

This is the car of the president
He is the boss of my brother
She's the girlfriend of Richard

But this is often considered incorrect by many people, so don't try to use it. It is safer if you always use the 'S for people.