When Windows is asked to connect to a network share, it always sends the
username (and domain name if there is one) and password of the currently
logged on user.  If this fails (the target computer doesn't "know" this
username and password - "authentication" fails), Windows displays a
credentials prompt for the user to supply a different username and password.

On Vista, the built-in user account called Administrator is "Disabled" by
default and can not be used to logon locally or connect to shares from
another computer; on XP it is "Enabled" by default.

When Windows XP or Vista is installed, a user account with the name
specified by the installer (or very first person to starts the computer), is
created  This user account is added to the local Administrators group.

If you are not sure what username you are logged on with for any reason
(e.g. you get logged on automatically):
1. click Start, Run
2. key
      cmd
    press Enter
(or, luanch Command Prompt via All Programs, Accessories etc.)
3. key
     set username
  press Enter

This command will respond with the username of the currently logged on user.

I'm not familiar with the LaCie ethernet mini disk, but the Manual for it
http://www.lacie.com/download/manual/ethernetdiskmini_en.pdf says that one
can use its built-in web site to configure "Users".  You might want to
create a "User" on the mini disk that has the same username and password
that you logon to your XP or Vista computer with.  This should avoid the
prompt for credentials you get when you connect to "shares" on it.