What Word forgot to tell you, however, is that their rules don't necessarily work well in the real world. Just because Word doesn't tell you that you used "their" wrong, doesn't mean you're right. In fact, Word very rarely correctly differentiates between homonyms or contraction and possessive forms of said homonyms.
Contrary to popular belief, there is a difference. And that difference is important. It's also fairly simple.
Homonyms are groups of words that sound the same, but have different meanings, and can include possessive and contraction forms of a given word (such as your/you're, or their/they're).
If you can rewrite (or read) a sentence with any of the following words, you're looking at a contraction:
If none of those words apply, you're typically not looking at a contraction. You're looking at a possessive pronoun, adjective, etc.
Your - possessive, as in: Your welcome was most appreciated.
You're - contraction, as in: You're welcome.
Its - possessive, as in: Its tentacles were huge.
It's - contraction, as in: It's almost bedtime.
Its/It's is a little tricky because most of us are used to adding an apostrophe when referring to someone's possessions (like that). Its is one very big exception to that rule. The only time an apostrophe is used is when you're using the word as a contraction. No exceptions. So, the claws might belong to it, but it is special, so we skip the apostrophe.
Whose - possessive, as in: Whose clothes are these?
Who's - contraction, as in: Who's in the bathroom?
Their - possessive, as in: Their house is on fire.
They're - contraction, as in: They're going to be in so much trouble.
There - refers to a place/location, as in: The dog is over there.
When in doubt, read it out.
If you're still not sure if you need the possessive or contraction form of a word, read the sentence out loud. If one of the contractions from the list above fit, you're looking at the contraction, not the possessive.
Contraction and possesive forms of homonyms aren't the only ones we frequently mess up. In fact, they're just a few of them. A few other common offenders, none of which are possessive or contractive, are:
To - preposition, as in: I want to go to the store.
Too - synonym for also or an indication of excessiveness, as in: I want to go, too. or: I have too many books.
Two - the spelling of the number, as in: Two heads are better than one.
Than - comparative, as in Her room is better than his room.
Then - time or sequence, as in Let's swim first, and then surf later.
Peak - summit, as in: We climbed the peak of the mountain.
Peek - glimpse, as in: She tried to peek at her gift.
Pique - stimulate, as in: He piqued my curiosity.
Rain - water, as in: Is it going to rain today?
Rein - a leather strap used to guide, as in: Please rein in the horse before he kicks someone.
Reign - period of occupation or governance, as in: The dictator's reign was full of abuses of power.
You can find a more comprehensive list of commonly mixed up homonyms here.
In the case of a homonym, one cannot substitute one word for another without changing the meaning of the sentence. There are very few, if any, exceptions to this rule.
Pretty simple, right?