Firefox automatically creates backups of your bookmarks and saves the last 15 backups for safekeeping. This article describes how to restore your bookmarks from the automatic backups that Firefox creates, how to save and restore your own bookmark backup files, and how to move your bookmarks to another computer.
You can use Firefox Sync to move your bookmarks from one computer to another.
Firefox Sync is the best way to keep your bookmarks (and other profile data) synchronized between all of the computers you use. See How do I set up Sync on my computer? for more information and instructions on setting it up.
You can also use a bookmark backup file from one computer and restore it on another computer. This is useful if you can't synchronize the two computers' bookmarks using Sync, for some reason.
The bookmark backup file can either be a manual backup (see above) or one of the automatic dated backups located inside the Firefox profile folder named bookmarkbackups folder. Place the bookmark backup file on your transfer media (for example a Flash drive) and copy it to the desktop (or any location) of the other computer. You can then restore the backup from the Firefox Library window, using the Choose File... option, as described in the Restoring from backups section above.