Assuming you backed your phone up in iTunes, the backup is intended to be used to restore all the contents of an iPhone onto another iPhone (or the same one if you need to restore from backup). If you get a new iPhone, then you can restore one of your earlier backups of the old phone onto your new phone.
The backed up info will be in iTunes. See http://support.apple.com/kb/HT2109. If you follow these steps, on step 4 you'll see a list of your previous backups, identified by date and time. You'd just select the last backup you did.
I don't know how you can extract just a piece of that backup, like just the contacts, although it might be possible. That might be a genius question. When you look at the actual files that are in the backup, it's incomprehensible.