There are really only two places you should use an apostrophe:
- possession (Jane's nose, the cat's pyjamas, the girls' books), and
- contractions (can't, won't, shan't; fo'c'sle).
And that's it.
Some would argue that you can use them with non-words (P's and Q's), and I'd suggest you should err on the side of caution and leave 'em out.
There are two places you should never ever use an apostrophe:
- plurals of regular words (known as the grocer's apostrophe)
- possessive personal pronouns (yours, hers, his, ours, theirs, its, whose).
Now, the word its and it's can be a little tricky if you're not paying attention. After all, its is possessive, right?
Here's the rule:
- if you could write it as it is or it has you can use the apostrophe. The apostrophe shows that you've left out a letter.
- if you made it plural and use their, then it's a possessive personal pronoun, and you want its. So, if it would be their house, then you use its house.
And yes, I do that quick mental substitution whenever I'm in doubt.
Oh, and if you're ever tempted to use her's, don't. After all, you'd never use hi's, would you?
And here's the Wikipedia page with more detail about special cases. For example, you can use it's house if you're writing in the early 1800s or earlier, otherwise nope, not happening.