My address book is on one computer, but can't be accessed by my other computers.

Do I understand right? You imported (thats downloaded ) your address book on to one comp. Do you want to repeat and import it again from the same source on to another comp? Or do you expect the address book to stay on the server?

richard.gg said

I imported my address book my hughes.net e-mail account on thunderbird successfully from sbcglobal.net. However, I cannot access my address book on any of my other computers. No problem with the e-mails themselves, just the address book.

I guess I need the address book on a server so all new added addresses will be available to all my computers.

Problem is to keep address book updated on server as that is controlled by the server-provider. Here is a link with several suggestions http://kb.mozillazine.org/Sharing_address_books