TweetMonkey is a Greasemonkey user script.
Firefox 3 or above with Greasemonkey 0.8 or above must be installed.（My testing environment is Firefox 3.013/Greasemonkey 0.8.3）
Greasemonkey 0.8 is a Firefox plugin and you can get it here.
Greasemonkey allows you to customize Firefox. Ask Dr. Google for details.
How to install
1. Access to the link below with Firefox 3
Install the Greasemonkey by following the instructions.
2. Click a monkey icon at the bottom right corner of Firefox to edit the TweetMonkey source code.
Put your twitter information on the lines of username/password at the beginning of the source code.
var username = '(your twitter account)';
var password = '(your password)';
3. It's done. Access to any web page to see if a Twitter input box will show up at the bottom right corner of the screen.
How to use
A Twitter input box will show up at the bottom right corner of the screen on any web page. You can enter a short URL by clicking on "Add short URL of this site" link. Tweet by clicking on "Update" button. You can use it just as on Twitter.
To tell you the truth, I usually use Google Chrome. I don't really use Firefox so often recently. I hear that Chrome can be customize in a similar way to Greasemonkey. Probably, I should give it a try on Chrome.
Any feedback will be greatly appreciated!!