It's and its. They're ubiquitous, and ubiquitously confused.
In theory, the rule that distinguishes the two is simple: it's means it is or it has. The apostrophe signals that something has been removed:
It's raining. [=It is raining.]
It's been raining since last night. [=It has been raining since last night.]
Meanwhile, its means "of or relating to it or itself":
Let the medicine do its job.
The door shuts on its own.
But this rule wouldn't have worked a few centuries ago. When it appeared with an s in the early 1600s, an apostrophe was involved, and the resulting it's meant "of or relating to itself," as in "a house with it's own little garden."
This apostrophe form of the possessive remained extremely common throughout the 17th century and was used by the likes of Thomas Jefferson and Jane Austen. The version without the apostrophe only became dominant in the 18th century—probably because it's was taking on a new role, replacing the contraction 'tis.
It would be simpler, of course, if there were only one form in use, and there's an argument to be made for using it's in all cases; 's serves both purposes just fine for nouns. In the cat's bowl it signals possession, and in the cat's sleeping it represents the contracted verb is.
But in current established English there is indeed a distinction between the two, and we recommend that you follow it: use it's only when you mean it is or it has, and drop the apostrophe everywhere else.