See these articles:

• Profiles - Where Firefox stores your bookmarks, passwords and other user data
  
• Managing Profiles
  
• Restore bookmarks, passwords, and data from an old Firefox profile

Thank you. I'll take a look.

You're welcome.

Assuming one of the articles answers your question, could you please click the "Solved It" button? This will help others searching for a solution to the same question.

Thanks.

I'm not sure if it solved things for me. I backed up my bookmarks to My Documents thinking I could download it to a CD, but I found Windows couldn't open it, I searched on the web and downloaded the recommended, but when I go back to open my Bookmarks document, I still can't open it and the circle starts over.

If I get the new computer, download Firefox to it, will I be able to access my existing bookmarks and the rest of my profile by signing in from the new computer?

You can move your Profile from one computer to another. That info is contained in this paragraph of the second article I linked to: Moving a Profile

Thanks, I think I managed to save the profile to a CD.

You're welcome.

Could you please click the "Solved It" button next to the post which solved the problem for you? This will help others searching for a solution to the same question.

Thanks.