So, let's begin, shall we?
There: This is a place. A specific location, sometimes to which you may point.
Example: "Where is the hawk?" she asked, her eyes shaded by the delicate curve of her hand.
"It's right there," he said, pointing at the bird circling above, a mere black speck against the heat of the sun.
Do you see how he is pointing at a specific place? There is the location of the hawk.
Their: This is a possessive adjective, of the third person. So, in other words, it's the plural of his or hers.
Example: The cup belongs to the Humperdink couple. It's theirs.
Example: Their eyes shone like glimmering lodestars in the dark of night, the only beacons in that endless sea of black.
In each case the word their denotes possession of the noun it modifies. One could say, The cup belongs to Prince Humperdink. It's his. When making this plural however, simply switch to their and you are good to go.
They're: This is a contraction of the words: they are. It is only ever a contraction. Never write, "Where does this Humperdink cup go?" she asked. "They're?"
If you do, know that a small cat somewhere is being smothered to death by dark avenging angels of the English grammar. And it will be all your fault. Not you're fault. Your fault.
Example: "Where are they?" Jake demanded, gazing anxiously from his seat on the porch for his daughter and her weasel of a Prom date.
"Calm down, Innis. They're just running a little late."
"They're forty-minutes late, Ethel! Where did I put my shotgun?"
Though the Humperdink couple barely made it through high school, they both employ proper use of the "they are" contraction. And so should you.