how can I back up my contact name list My contacts on my IPhone 4s was lost. If I backed up my phone to the computer, will the contacts also be saved? where can I find that or how can I restore my contact list?
Yes, contacts are saved in iPhone Backups, or in iCloud if you have that enabled.

This Apple support page illustrates restoring backups. http://support.apple.com/kb/ht1766

For restoring computer backups:


Connect your iOS device to the computer that has your backup
Make sure this computer has the latest version of iTunes
Choose File > Devices > Restore from Back up
If you're using iTunes 10.7 or earlier, right-click the device from the list and choose Restore from Backup.