Terms
Possessive: When the subject owns something
Plural: Multiple of anything.
Possessive plural: When there are multiple things that own something
Contraction: When letters are cut out and two words are mashed together.

Examples
Possessive:
If the dog has a ball [ownership]:
It is the dog's ball.

Contraction:
It is really hot today:
It's really hot today.

Plural:
If the dog is playing with another one:
There are two dogs.

Possessive plural:
If the two babies both have dummies:
They are the babies' dummies.

If a toy belongs to both of the babies:
It is the babies' toy.

Tricksters
It's/Its
I find the best way to work out whether to use it's or its is to check whether it is a contraction [mashed together] or possessive [ownership].

- You only ever use it's if it is a contraction. Remember that, and you're set for life. If you can remember this, you don't even have to remember the following rule.
- You only ever use its if it is possessive.

It's very hot today.
The dog chases its ball.

On Names Ending in "S"
It is grammatically acceptable to demonstrate possession with a name ending in "S" by either placing an ['s] on the end, or simply placing an apostrophe after the "S".

eg.
James's ship.
James' ship.

Short Rant On Names Ending in "S"
I passionately support the first option (full name, ['s]). Why? Because in the first option, the name is treated like all other names.
The reason I detest the second option is that the name is then treated like a possessive plural. And it's not. There is only one James.