Their | there | they're

All three are pronounced the same when spoken, and this causes students to confuse them when writing.

There

There means place or direction. See how it contains the word here in it (which can help you to remember its connection with place).

The book had been sitting there for some time.

There are only two certainties in life: death and taxes.

There's no end in sight with the issue of corruption in Russia

There can also have a more abstract meaning when used in conjunction with is / are to suggest the existence of something. This is especially true if the subject of the sentence appears after the verb. The word there behaves as a kind of dummy subject placeholder. Look at the following example:

There is a stranger hiding in those bushes.
(Instead of: "A stranger is hiding in those bushes.")

Grammarians have different theories about why the English language provides this option. Our favourite theory is that the sentence using There is.... shifts the focus of the sentence from a general assertion about the nature of the subject, to that of a specific instance requiring particular attention. The example above goes from a statement about any ubiquitous stranger (randomly hiding in bushes) to being more about the specific location of a particular (ominous) stranger (Hardegree, 2015).

Reference:
Hardegree, G. (2015). There insertion [Course handout]. Formal Semantics, 2015-04.21.

Their is a possessive adjective and means only one thing: that something belongs to them...

Although the students wanted to stage a protest against the University's academic board, it was not in their best interest to miss the exam.

The twins explained to the media that their mother had been a great influence on them, and instrumental in their success.

They're is a contraction of they are, and that is the only meaning it has.
Therefore, you would not use it in any of your assignments because contractions are not sufficiently formal for academic writing.

The security guard saw them stealing, and now they're being held until the police arrive.

Tomorrow, they're being removed from the village for their own safety.