While a many of you already know that these words are different -- it's not a lot of help to have that pointed out without a trick to assist you in using them correctly.
That's where the Writers and Polishers come in...
There -- is the opposite of 'here.' Conveniently, it also contains the word 'here.' So, if you mean the opposite of 'here,' slap a T on the front, and you're all good.
They're -- is a contraction (mash-up) of 'they are.' The apostrophe takes the place of the space and the A. If you mean to give information about a group of two or more people that doesn't include you, this is the one you want.
But, to be thorough:
Their -- shows possession. If something (an object, an idea...whatever) is owned or held by a group of two or more people that doesn't include you, describe it as theirs. The test here would be if you were talking about a group that DID include you, could you replace 'their' in the sentence with 'our' and it would still make sense? If so, 'their' is the choice you want.
Marsha, Jan and Cindy usually sided with their mom when there was a disagreement with the men in the house.
If you imagine for a minute that you are Cindy, the sentence would make sense reading, "Marsha, Jan and I usually sided with our mom when there was...."
This is a slightly more complicated test, but not really difficult once you make a habit of stopping to check that you've chosen the correct homophone (words that sound the same) when writing there/their/they're.