As an exhibition at the Jewish Museum in Berlin from 2005 explained, the concept goes back over a century to a time when increasingly assimilated German Jews appropriated Christmas celebrations in their own secular manner. (The original term was Weihnukkah, of which Chrismukkah is just an anglicization.)
In any event, greetings of the season on whichever holiday(s) you happen to be celebrating.
In recent decades, the term has become respectable--half-serious and half-humorous--and taken on a life of its own.
This image of the menorah evolving into the Christmas tree comes from a postcard sold by the Museum, and the original intent was critical rather than celebratory. The caption reads:
"Darwinian: Zionist caricature on assimilation, from the periodical, 'Schlemiel' (1904)"
That these issues still arouse strong sentiments can be seen from this rather less subtle blog entry by Jeremy Cardash and its responses at the Jerusalem Post.