It's "its," not "it's" - most of the time

An extremely prevalent word usage error these days is that of "it's," with an apostrophe between the t and the s, when the correct word is "its."  Yes, both are proper words, but "it's" is a contraction of either "it is" (as in, "It's a beautiful day today") or "it has" ("It's been a beautiful day") - and those are its only meanings.  When possession is indicated, the word to use is the possessive pronoun "its," as in the last phrase of the previous sentence.  (This example uses both words: "The painting has lost its luster; it's not as bright as it used to be.")
Of course, I've seen the trend reversed as well, when the apostrophe is incorrectly omitted where it's required.  But that's another blog post.

Possessive pronouns, such as "its," "whose," "hers," "ours," "yours," and "theirs," do not require apostrophes.  Spellcheckers probably won't flag either of the first two, because "it's" and "who's" are proper words (there are, however, no such words as "her's," "our's," "your's," or "their's").  This can be confusing because most other nouns and all proper names do use the apostrophe-"s" combination to indicate posession ("Whose bicycle is that?"  "That is Sally's bicycle.").  But, as we all know, there are exceptions to nearly every rule.  As a consequence, it's up to human beings to memorize the rules of grammar pertaining to the usage of "it's," "its," "who's," and "whose," or to use an on- or offline reference when unsure.  (There are links to a few good online ones under the Resources heading in the sidebar of this blog.)

To summarize:

It's = it is or it has (contraction)
Its = belonging to it (possessive pronoun)
  Incorrect usage: "The snake is shedding it's skin."
  Correct usage:  "The snake is shedding its skin."  "It's going to be a good day."

Who's = who is or who has (contraction)
Whose = belonging to whom (possessive pronoun)
  Incorrect usage: "Who's bicycle is that?"  "There's the man who's eyes are two different colors."
  Correct usage: "Whose bicycle is that?"  "There's the man whose eyes are two different colors."  "Who's going to the store?"

Her's, their's, our's, your's = these words do not exist

See also Common Errors in English Usage - one of the resources listed in my blog's sidebar, and a good one.