The screenshot has not been approved by moderators at this time but it sounds like you are describing a "favicon".
The most simple method is to create a standard icon file, name it 'favicon.ico' and upload to the root directory. Almost all modern browsers would automatically recognize it if accessible at yourdomain.com/favicon.ico
In addition to that you can create slightly higher resolution .png or .gif as well as alternate versions for mobile devices using link tags in the head of the html documents. Here are a couple articles on the subject: