THERE is used to say that something exist in a particular place. The word "there" can be used in singular or plural. It depends on word that follows it.
There is my cat. There are my two cats.
THEIR is a possessive adjective from subject THEY. It always used before noun. For better memorization look at the table
you-your you-your There are my two cats. Their fur is dirty.
THEY'RE it's a contraction. ( Two words that are pushed together =THEY ARE )
They're students. They're learning. They're sad today.
The singular form for the same sentences is:
I'm a student. I'm learning. I'm sad today. OR
You're a student. You're learning.You're sad today. OR
He's a student. He's learning. He's sad today.