The contraction it's stands for either it is or it has. The word its, however, is a possessive pronoun that means "of it." The apostrophe simply marks the missing letter and does not imply possession in this case.
The confusion arises because apostrophes do indicate possession for nouns, as in the dog's collar. As a result, writers are often tempted to apply the same rule to it. A simple test is to substitute it's with it is or it has. If the substitution makes sense, use it's. If it does not, use its. For example, it's raining = it is raining ✓, while the cat used it's paw would become the cat used it is paw ✗, so the correct form is the cat used its paw.
For more grammar tips, see our guide on passive voice vs. active voice. Quetext's Grammar Checker catches its/it's errors automatically.