I have 2 Vista computers, the desktop has a printer, and is run under
administrator with a password.
From the Laptop, if I explore the network, and put in the password, AND
CHECK REMEMBER PASSWORD the laptop will print across the wireless network to
my desktop printer for that session. The next day, after the laptop has been
shutoff and started, the saved password is gone, the box is unchecked, and it
won't print, until I explore the network and sign into the desktop.
The desktop usually does not get restarted unless some update requires it.

How do I save my login on the laptop so the printer will automatically be
connected across the network?

You can try these steps but it may not work in Vista Home Premium. I've read where Home Premium doesn't support this so there's not really an easy way around it.

1 - Reboot the computer
2 - After you log it will say your mapped drive couldn't be connected and stuff.... ok
3 - Go into Control Pannel, User Accounts, Network Passwords, and add the \\server\share, with the username & password you want
4 - Open My Computer, and access your mapped drive
5 - Reboot the computer