These instructions were written for Thunderbird but also apply to the Mozilla Suite. The main difference is that Tools -> Account Settings is replaced by Edit -> Mail & Newsgroup Account Settings.
Gmail supports webmail, POP, and IMAP accounts. They all access the same mailbox. Gmail also provides a SMTP server that you can use to send messages. To add an account in Thunderbird you need to add either a POP or IMAP account, enable it in Gmail using a browser, and then configure the account to use the SMTP server.
If you want to read your messages offline its frequently better to use a POP account rather than a IMAP account. IMAP stores messages in remote folders. You can configure a IMAP account to mirror folders locally using Tools -> Account Settings -> Offline & Disk Space -> “Select folders for offline use” but its more suitable for advanced users.
Gmail is rebranded as Google Mail in Germany, Austria and the United Kingdom.