Custom Badges are a major part of the income for Furry Artists these days. While they bring original art for sale in the Art Show, and art-quality prints (usually limited-edition) for sale, they have come to rely on Sketchbook and Badge Art commissions for their "walking around" money at the conventions. THe custom badges can be commissioned from many artists in advance, for delivery at the convention; see the web-page of your favorite artist for details about that, or check out FurAffinity or whichever favored web archive your favorite artist uses to post new work.
Custom badges are essentially Greeting-style nametags, with your fannish nickname or Game Identity, illustrated. The badge may be colored and laminated, or colored and given to you unprotected, except for a coat of an artists' "fixative", to keep graphite or pastel or colored pencil from smearing, and then you put it in a badge-holder of some sort.
A couple of tips: After you receive your badge - and this also applies to Sketches in a sketchbook - it is considered good form to scan the art image, and send the scan back to the artist, so that he or she has a copy for their portfolio. (If you commission the artist beforehand, this will be done by the artist before they deliver it to you).
If you receive the badge art and it is "raw" or unprotected, you may wish to copy it, and put the copy in your badgeholder, and keep the original in a safe place; That way you can easily make another, if the one you wear gets lost or trashed in some accident.
As mentioned above - the custom badge is a friendly ID tag, to let other Furry Fans know who you are. The OFFICIAL EuroFurence Badge is your convention "Credential", and is necessary to access the programming at the convention - and last year, it doubled as your hotel room key, too!