A favicon, short for "favorites icon," is a small graphic you can associate with your website for display in browsers like Internet Explorer, Firefox, Mozilla, Opera, Safari, iCab, and AOL Explorer. This icon appears in several places: in the browser's Location field, next to the site's name in a bookmarks list, with the site's title in a tab (if the browser uses tabs), and next to the site's feed as shown in many feed readers.
What does it look like?
If you don't have a favicon already, a search for favicon.ico reveals a number of sites that explain how to create an ICO file and how to host it on your website so that browsers and feed readers can locate it.
The problem with favicons and FeedBurner feeds stems from how different feed readers locate the favicon. Many use the address of the site provided in the top-level <link> tag of the feed to locate it (in this case, they will find yours, which works well with NetNewsWire and My Yahoo! right now, for example). Others, especially some desktop feed readers like FeedDemon, use the address of the feed itself and don't look back to the website (which means FeedBurner's favicon ends up getting the nod).
There is currently no solution for this issue; FeedBurner may offer one in the future; this topic will be updated to reflect any new capabilities.