Note: You are currently viewing documentation for Moodle 1.9. Up-to-date documentation for the latest stable version is available here: Favicon.
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 eg /theme/standard/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.)
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.