The apostrophes in contractions take the place of the letters that get knocked off when two words are combined into one, thus "I am" becomes "I'm," and "it is" becomes "it's." In contrast, possessive pronouns inherently convey possession without any apostrophe. Just like we write "his" and not "hi's" or "his'" for belonging to him, we write "its" (with no apostrophe) for belonging to it.
Here's the list of pronouns in contraction and possessive form:
Contraction Possessive Pronoun
From the list above, people tend to confuse the ones that sound alike: it's/its and they're/their. Interesting enough, I don't really see "you're" mistaken for "your," even though both words sound exactly the same, The thing to remember is that just as you would not write "you're" or the nonexistent "you'r" for the word that mean belonging to you, you cannot write "it's" for the word that means belonging to it, no matter how common the error.
The same holds true for "their," which is not only confused with the contraction "they're" but even with the third homonym for the grouping, "there."
Note: for the rules of apostrophe usage for plural forms, see http://www.dailywritingtips.com/when-to-form-a-plural-with-an-apostrophe/