Did you add the user's domain login to SQL Server? Builtin\Administrators is added by default during SQL Server installation. You'll have to add any other individuals or groups that you want to access SQL Server. See "Logins [SQL Server]" in BooksOnLine.
I don't use Express, but I think it comes with an version of SQL Server Management Studio which you can use to add logins. In Object Explorer, expand Security, right-click on logins, and select "New Login...". Once you've added the login, you can grant database access and other permissions.