Flensburg does not have the most beautiful Christmas market in the world - basically they have booths along the main shopping street.
BUT they sell some unique drinks there, called Punsch. It's like any other Glühwein you get in Germany, but you can add some more rum, and you can add raisins, special sugar and other spices, just to your taste.
Many of the booths also have a wide selection of hot chocolate with a shot of rum or other stuff.
And to eat, you should try Grünkohl and Grünkohl soup. It's a very health winter vegetable, doesn't look very nice but the taste is great. Typical food for the north of Germany.
Das Auge isst nicht mit, aber das Ohr trinkt mit! (sorry, untranslateable!)
Some of the backyards have great little christmas markets as well, especially around the Rote Straße.
Flensburg is Germany's most northern city, just a few minutes away from Denmark. It used to belong to Denmark, a long time ago, there are Danish schools and newspapers in the city and lots of visitors from across the border.
Flensburg is full of suprises: there are a lot of hills, and steep streets in the city; rum is of huge importance; not that many fish restaurants as I had expected; a very long shopping street, beautiful old merchants' houses, nothing was destroyed during the last wars, terrific backyards with galleries, little shops and cafes.
And Flensburg is full of what you expect: Flensburger beer everywhere, harbour right in the city center, seagulls, people talking with a funny accent saying "moin" at any time of the day, and a huge market square like in the "werner beinhart" movie.
A relaxed athmosphere, friendly people, close to lots of beaches, and the best thing: my brother is studying here now and I'll have free accommodation for the coming years :-)