Do not confuse they're, there and their. A mistake involving these constitutes a grammatical howler.
They're
They're is a shortened version of 'they are'. (The apostrophe replaces the letter
a.) Only use they're if you can substitute it with 'they are'.
Examples:
They're not leaving on Saturday at all.

("They are not leaving..." < sounds ok)

This is wrong - should be their
(office sign by sink)
I cannot believe they're from Wigan.

("...believe they are from Wigan." < sounds ok)
More than 20 people left they're coats in the cloakroom.

("...20 people left they are coats in the cloakroom" < nonsense)
Why do you listen to them? They're unqualified.

My doctor gave me two weeks to live. I hope they're in
August. (Ronnie Shakes)
Things are only impossible until they're not. (Jean-Luc Picard)
TheirTheir is used to show possession. It is just like my,
your, his, her, its and our.
(These are called 'possessive adjectives'.) Here is a little trick: use the word
'our' instead of 'their'. If the sentence still makes sense, then 'their' is almost certainly correct.
This trick works because 'our' and 'their' are both possessive adjectives used for
plurals.
Examples:
Can you show the guests to their cabins?

("..show the guests to our cabins" < sounds ok; "their" is correct)
I have seen their footprints before.

("I have seen our footprints before." < sounds ok; "their" is correct)
Their all leaving.

("Our all leaving." < nonsense; "their" is wrong; should be "they're")
Their less likely to cause offence.
Experience is the name every one gives to their mistakes.
(Oscar Wilde)
There
The word there is similar to the word 'here' in that it represents a place. It has two main uses: It is a specified place (like in the first example below) and an unspecified place (like in the second
example). Also, like in the second and third examples, the word 'there' can be used to show that something exists.
Examples:
The Germans are over there.

(specified place)
There are two apples.

(unspecified place - two apples exist)
There are two apples left in the fruit bowl.

(two apples exist; place specified later in the sentence - i.e., in the fruit bowl)
They're their coats over there.

("They're" = they are (explained above) / "their coats" -
possession (explained above) / "there" - place)
Simon looked up and repeated his opening line: "They're unqualified and their
opinions counted for nothing while they were their." 
(last "their" should be "there")
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NOT IN FORMAL LETTERS
Never use words like they're and doesn't in formal letters. Always expand them to
they are and does not. (See lesson Apostrophes.)
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