Someone who's long-winded takes way too long to tell a story or give a speech. Once your long-winded history teacher gets started on an anecdote about the Civil War, everyone in the class sighs deeply.
A long-winded explanation is one that uses many words when just a few would do. And your grandfather's long-winded toasts before Thanksgiving dinner inevitably end in everyone's food getting cold before they can start eating it. An older meaning of this word was "able to do something for a long time without losing one's breath," but the modern definition dates from the 1580s, when it specifically meant "given to lengthy speeches."