If your new computer came with a trial version of Office 2003
preinstalled on the hard drive by the computer manufacturer, then
you should have uninstalled the trial version before installing
your full version. At this point, uninstall the full version,
uninstall the trial version, and reinstall the full version. See
http://support.microsoft.com/?kbid=836178 "You receive an
'invalid product key' error message when you start an Office 2003
program after you install Office Professional Edition 2003" for
more information.
If you have a retail copy of Office 2003 Professional, then you
can install it on one desktop computer and one laptop computer
both owned by you. If you have an OEM copy of Office 2003
Professional (it came preinstalled on your old computer by the
computer manufacturer), then you are not allowed to transfer the
software to your new computer.