The small icon shown in the browsers in front of the URL is called a favicon.
You can use your favicon by replacing the default favicon.ico icon with your own. Typically, the favicon.ico file is located in your theme's folder, for example, in ../theme/(theme name)/pix/favicon.ico. Make sure to keep your favicon.ico file named "favicon.ico." If you are not using a standard theme, you will have to find the "favicon.ico" file that your custom theme uses.
- The change may not be noticeable 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. Site Administration > 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. You might also need to clear your Moodle's cache by going to Site Administration > Development > Purge caches > Purge all caches.
This Wikipedia article explains how favicons work 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.