This song details nightlife on Cyprus Avenue, which is identified in the song's first line ("down on Cyprus Avenue"). Cyprus Avenue is a real place in Van Morrison's hometown of Belfast, Ireland. It's also the title and subject of another Astral Weeks song
There are a few characters in this song, but they all revolve around Madame George, who gambles, parties, and possesses something that requires tossing out a window during a police raid.
For most listeners, it seems obvious that Madame George is a drag queen. Morrison himself has denied this, explaining it's just a mishmosh of multiple people he'd known.
Still, many (including legendary rock journalist Lester Bangs) have seriously questioned Morrison's claim. In addition to the obvious juxtaposition of the feminine identifier "madame" (basically the French equivalent of "Mrs.") with the man's name "George," Morrison sings:In the corner playing dominoes in drag
The one and only Madame George
It's also possible that it's simply a woman named George, as throughout the song the character is referred to as "she" and "her." Then again, it may simply be a show of respect to a transsexual's preferred gender.
Whatever the character's sexual identity, he or she comes across as a goodhearted ne'er-do-well who also carries a lot of sadness.