The printer has a share name and is already setup on the laptop (computer B). When I print from the laptop, while selecting that printer, it shows that the document has printed. The problem is just that nothing comes out of the printer. Any other suggestions?
Thanks.

Did you give the priner a "Share Name".  You must install the printer on Computer B by using the Add A Priner Wizard.  Add the priner by using this method:

 

\\computerA\sharedprinter  where Computer A is the name of the Computer it is phyically attached to and shared printer is the Share name you gave the printer

Yes, I can print from the home computer. I'm fairly certain that the drive is up to date. How would I check that though?

Can you print from the "home" computer?  If might be a driver issue with the networked machine.  Check to see if there is a more current driver available than what you are using.

Go to the maker's website and search for the versions available.  I would go the Event Viewer on the home machine and see what errors are there associated to the printer and vice versa on the shared machine.

Downloaded updates, but the problem persists. Any other ideas?

You are now getting into a permissions issue.  Right click on the printer --> properties --> security tab.  Check to see if the usser you're logged on as has permissions to print to this printer.