Apostrophes are part of a word to indicate possessive case, contractions, or omitted letters.

The apostrophe always goes after the noun who owns the object.

• The kite's string was knotted and frayed.
• Frank's bedroom was very messy.

This is true, even if the noun is plural.

• The childrens' playground was muddy after the rain.
• The men's bowling league saw record participation over the summer.

If the noun ends in an 's', it depends on the situation.  The easiest rule of thumb is to write what sounds natural to say.

• Bridget Jones's Diary was released in 2001.
• The carpenters' union is one of the largest building trade unions in the country.

A contraction is a shortened version of spoken and written forms of a words or syllables which is created by omitting certain letters.

• I will -> I'll
• They have -> They've
• I had -> I'd

Numbers can be shortened by adding an apostrophe in place of the omitted number.

• Class of '11
• Back in the '90s

An apostrophe and s are also used to form the plural of letters, numbers, and words referring to words

• Do you spell professor with two f's?
• There are six nine's in my phone number.

Time Periods

• The 1990s

After Numbers

• MP3s

After Symbols

• *s
• #s

After Abbreviations

• CDs
• DVDs

With Possessive Pronouns  (its, hers, his, theirs, my, mine, ours, yours, whose)

• The cat licked its paws.

An easy trick is to remember that the pronoun is possessive, so it wants nothing between itself and the s--aka, no apostrophe!