Generally, MS Office is rather good in having multiple versions on the same machine. So it might very well work.
Office 2003 certainly allows you to install only one product. But as the product key is activated on your desktop, it might refuse activation on the laptop without you buying a new license. That depends, of course, on the license you have. As far as I know, Home and Student edition allows for three installs. But we don't know what version you have, and what is possible and allowed with it.
If it doesn't work satisfactorily, you can try to uninstall Word 2007 from the Office 2007 CD, then reinstall Word 2003 for the Office 2003 CD. It's not 100% to work, but I'd give it a 99% chance.
The one thing not clear in your post: "I need to use Word 2003". Why can't you use Word 2007? Or OpenOffice 3.0?
I just bought a laptop with Microsoft Office 2007 Small Business. Right now, I need to use Word 2003. I have the Microsoft Office 2003 CD from my Desktop PC. Can I install only Word 2003 from the CD or does it require you to install all of the products; Outlook, Excel, etc.?
If I can install Word 2003 only, will the install have a problem with Word 2007 being on the laptop? What other things and concerns do I need to be aware of?