It's very common to choose the wrong words when trying to write possesive pronouns. But before I end up getting too deep in jargon, let's just take a look at some examples:

Possessive pronouns
Imymemine
youyouryouyours
he/she/ithis/her/itshim/her/ithis/hers/its
weourusours
theytheirthemtheirs

So like I said, I don't want to get technical about defining this stuff: my point is that none of the above words have apostrophes.

Most of the confusion seems to arise from its vs. it's, and this largely seems to stem from possessives of the form Alice's, Bob's Your sister's, etc. So it's easy to see why it's so tempting to write it's to mean belonging to it. But unfortunately, English is consistently inconsistent, and it's can only ever mean it is.

Contractions
I'mI am
you'reyou are
he'she is
she'sshe is
it'sit is
we'rewe are
they'rethey are

All of these words are contractions: their apostrophes stand in for deleted letters. It's unfortunate that the apostrophe is used for Alice's, etc., since apart from that, it's only ever used to indicate missing letters in contractions like these. More examples:

Contractions
can'tcannot
don'tdo not
won'twill not
who'swho is

So every time you write an apostrophe, think about whether you really need it -- does it stand in for missing letters in a contraction, such as in the above examples? Or is it being used to form a noun's possessive? Then go right ahead. But if it's for a possessive pronoun (its, your, etc.) or a plural noun (cars, houses, walruses, buses), then you don't need it.