The small icon shown in the browsers in front of the URL is called a favicon.
To create your own replace the file favicon.ico in your theme directory. For example in /theme/standard/pix/favicon.ico with your own icon. If you are not using the standard theme you will have to do this in the folder corresponding to the theme you have chosen.
- The change may not be noticable because the previous image may be in the server or browser cache.
- Tip: Favicon is not changing on your browser? Moodle cached the old one. Settings > Appearance > Themes > Theme settings > Theme designer mode check box and save. Then hit your browser's refresh button. You might want to uncheck the box after it works.
This Wikipedia article explains how it works and has some useful links to software you can use to create one. It also mentions some online utilities that can be used to convert pictures to the right format for use as a favicon.