If you are using POP3 with your Gmail account then setting it up in Thunderbird is a breeze – just select the Gmail option and enter your username and password. Thunderbird will do the rest.
However if you choose the IMAP option you will have to manually set up the account with all its individual settings – an inconvenient process.
Gmail IMAP Account Setup is a Thunderbird addon that automates this process of setting up your Gmail and Google Apps IMAP account in Thunderbird. Just install the addon and the next time you create a new email account you will be shown the option to setup an IMAP account [previously not shown]. Select it and Thunderbird will take it from there and setup the account for you.
Google Apps is a service provided by Google where you can use Gmail with your own domain name (example abc@xyz.com). Here though you are using a Gmail account, your email id will not show @gmail.com but your choice of domain name @xyz.com
If you want to set up Gmail with your own unique domain name, you may take a look at this post:
Creating your own, unique email id - Google Apps
If Microsoft you worship:
Creating your own, unique email id - Microsoft's Live Domain
Godspeed