I recently activated my home network with two computers. The printer is shared but I can not do back up from the computer not directly connected to the external drive. I am using Windows Live Onecare to do backups. I enter \\computer name connected to drive\backup.
It replies that I need write permission.My OS is XP home edition.
Please,any advice would be appreciated.

Your external drive may not be shared. Go to the machine that has the external drive attached to it and Right click on the drive and choose "share" and then follow along the menu options to name it and allow permissions.

Make sure you have administrator ability on both computers it may be a permission issue.