Use its when something belongs to a thing, animal, or idea, not a person. The city is famous for its markets. If the word you're choosing can stand for it is, you need it's instead, as in It's late.
### Situations where *its* fits
Describing an object: use its when the thing owns something. The laptop lost its charger at the airport. The charger belongs to the laptop, so the plain word with no apostrophe works.
Talking about an animal: use its when the animal has something. The cat licked its paws clean. You're not calling the cat he or she, so its keeps it neutral.
Writing about a company or place: use its when a singular organization or place is the possessor. The company moved its offices downtown. The offices belong to one company. If you wrote about several companies, you'd switch to their instead.
Introducing an idea or story element: use its for abstract possessors too. The novel found its audience slowly. The audience is associated with the novel, not with a person.
Contrasting case: reach for it's only when you could say it is or it has in that spot. It's raining again holds up as It is raining again, so the contraction is the right word there.
The sixty-second check before you hit send
You've written your sentence and now you're hesitating. Try this: read the phrase with it is or it has in place of the word you chose. If the sentence still makes sense, you need it's. If it collapses, you want its.
The dog wagged its tail. → The dog wagged it is tail? No. So its is right here.
It's been a long week. → It has been a long week? Yes. So the apostrophe belongs.
One caveat: this is a rule of thumb for choosing between the two spellings, not a universal grammar test. If a sentence feels odd under substitution for some other reason, check the whole structure rather than trusting the test alone.