A favicon (short for favorites icon), also known as a shortcut icon, website icon, URL icon, or bookmark icon is a 16×16, 32×32 or 64 x64 pixel square icon associated with a particular website or web page. If you look at your browser information header for this site you will see a tiny spider. That is the favicon for this site.
Your favicon is your mini-logo - it appears on your browser page next to the site name each time you visit the site and if you have saved the link on your bookmarks tool bar it serves as a visual cue to the site you want to visit.
All of the main browsers support a 64 x 64 .ico nowadays, but often it is better to go for a bold simple style.
Once you have established your logo it is sometimes possible to use an aspect of that in your favicon. But simplicity is the key.