English has three of these that sound alike, and two of them cause most of the trouble. "There" and "their" get swapped constantly in email and chat, and unlike most spelling slips this one gets noticed — it is the error people quietly judge. The good news is that each word contains a smaller word that gives it away, and once you see that you stop having to think about it.
An adverb of place, or the word that starts a sentence about existence: "there is," "there are."
A possessive determiner. It marks something as belonging to a group of people already mentioned.
There, their and they're sound identical, so the ear is no help. Each does a completely different job.
There points to a place (put it over there) or says that something exists (there is a problem). Their shows that something belongs to more than one person, or to one person whose pronoun is they: their car, their idea. They're is a contraction of "they are" and nothing else.
Two quick tests settle almost every case. If you can expand the word to "they are", it is they're. If you can swap in "our" and the sentence still makes sense, it is their. Everything else is there.
Their going to be late for the meeting.
✓They're going to be late for the meeting.
This means "they are going", so the contraction: they're.
The team left there laptops in the office.
✓The team left their laptops in the office.
The laptops belong to the team. "Our laptops" makes sense, so their.
Their are three reasons we should wait.
✓There are three reasons we should wait.
The sentence says the reasons exist, which is there.
Quick way to remember
Look for the smaller word hiding inside. "There" contains "here" — both are about place. "Their" contains "heir" — and an heir inherits, which is possession. The third one, "they're," is just "they are" squeezed together.
Is "there's" correct?
Yes, as a contraction of "there is" or "there has": "there's a problem". In formal writing, "there are" is preferred before a plural: "there are two problems", not "there's two problems".
Can "their" be used for one person?
Yes. Singular they, and with it their, is standard in modern English when a person's gender is unknown or they use they: "someone left their umbrella."