A badge description is the description of a badge, explaining why it is awarded.
All badges on the Stack Exchange site have a description. The description of a badge explains the condition for awarding it.
The short description of a badge can be found on the Badges page.
Every Stack Exchange site and every Meta has its own Badges page, but the descriptions are the same across the network.
A more extensive description of each badge is found in the List of all badges with full descriptions.