Apostrophes

Apostrohes are used for two different purposes - what are they?

What do we mean by omission?

What do we mean by possession?

Corret or Incorrect?
mighth've

Correct or Incorrect?
should'ave

Correct or Incorrect?
that'd

Correct or Incorrect
Its time to leave for the party.

Correct or Incorrect?
On Monday its going to be a lovely day.

Correct or Incorrect?
Madrid is well known for it's art galleries.

Correct or Incorrect?
I dont think it's a good idea to walk home alone.

Correct or Incorrect?
The childrens books were in the cupboard.

Correct or Incorrect?
The car's are red.

Correct or Incorrect?
The player's boots were filthy.