I live in Pittsburgh, PA and I have always grown up eating pork and sauerkraut for my dinner on New Years Day. My grandma always said that it was what she ate on New Years Day growing up, so she continued the tradition with her family. My mom now does it and I can't imagine having a New Years without it! Eating this meal is supposed to bring you good luck for the upcoming year.
A lot of people hate sauerkraut because it is so bitter. The trick (if anyone wants to try it) is to dump the sauerkraut out of the can/bag into a strainer, rinse it with cold water, shake that out, rinse it again a few more times. That takes the horrible bitterness out of it and it actually tastes really good.
We usually cook it on the stove with the pork and also some hot dogs, but a crock pot works really well, too!
I think this originated from the Pennsylvania Dutch.
Edit -- Here is what the PA Dutch supposedly thought about it - "Pennsylvania Dutch superstition says that eating pork brings good luck because a pig roots forward to look for its food, while chickens scratch backward and cows stand still."
My family puts a can of plain cooked tomatoes in the sauerkraut along with the pork. Tomatoes give it a smoother flavor.