Q: how do I transfer my phone contacts from an Apple I phone 3 to my new Apple I phone 4S?

First, backup the old phone using iTunes on your computer.  Then, when you initialize the new phone, use the same computer.  There will be an option "restore from backup" or something like that, which will put the backed up information onto the new phone.

Hi Gerald. Your answer just answered my question. However, I still want to know what is the difference between synching and backing up or restoring backup. Hope you can help. Thanks, Ana

When you sync, only the changes since the last time are sent.  For example, if you bought a new song in iTunes, then when you sync that song will be sent to your iPhone also.  Or you added a new contact on your iPhone, when you sync that contact will be sent to your computer.

Backing up sends everything, not just what has changed.  So it usually takes much longer than a sync.  But it is good to have a backup stored on your computyer in case your iPhone is lost or damaged, so you can get your replacement iPhone to match the lost one.  (Or even, as in your question, you get a new iPhone.)

Since my original reply a year and a half ago, there is a new possibility:  You can use iCloud for backups.  Theoretically you never need to connect to a computer by USB cable, just do everything by wifi or G3 or whatever. 

(I say theoretically, because I do my backups with my computer... I'm not used to trusting the cloud with my only backup.  But in fact I do know people with an iPad and no computer, who manage with public wifi connections.)