You should use an apostrophe to form the possessive case of a noun or to show that you have left out letters in a contraction. Note that you should not generally use contractions in formal, academic writing.
The convertible's engine has finally died. (The noun "convertible's" is in the possessive case)
I haven't seen my roommate for two weeks. (The verb "haven't" is a contraction of "have not")
To form the possessive of a plural noun ending in "s," simply place an apostrophe after the "s."
He has his three sons' futures in mind.
In many suburbs, the houses' designs are too much alike.
Possessive pronouns -- for example, "hers," "yours," and "theirs" -- do not take apostrophes. This is the case for the possessive pronoun "its" as well: when you write "it's" with an apostrophe, you are writing a contraction for "it is."
The spaceship landed hard, damaging its radar receiver. ("its" is the possessive pronoun)
It's your mother on the phone. ("it's" is the contraction of "it is")