Applications that have a "degraded" mode generally offer access to a limited set of features due to an internal or external problem. For example, imagine that your application connects to two different data sources. If one of these fails it could still be possible for it to continue running but to disable all visual (or/and backend) features related to the unavailable resource.
Concrete example: your application has two data sources, in one of them you store persons, in the other you store accounts. Even if the account database fails, if the application has a degraded mode, you should still be able to create new persons, albeit not interact with their accounts.
As to the user interface, this generally means that you clearly indicate to the user that some features are disabled, and display some warning message so that he is not left out in the dark as to why something is unavailable.
Also, there is a Wikipedia entry related to fault-tolerant systems.