For special occasions -- but especially during the New Year's holidays -- people in Japan eat mochi. It's glutinous rice that is cooked and then while still piping hot pounded into a sticky (and I mean sticky!) dough. And from there it becomes any number of wonderful treats.
Traditionally and still in a lot of places it was pounded like this:
Now, though, they have special rice cookers with built in mochi makers that do most of the work. Even more popular (and easier by far) is just having someone else do the work and buying it at your local supermarket. Right before the New Year's holidays they have people selling fresh mochi in supermarkets all over the place. You can buy the dried kind, too, (that's sold year round) but it's not nearly as good.
So after the mochi has been pounded to within an inch of its life it is rolled out and cut into squares. Or sometimes while still soft rolled into balls. They are then stored in a cool place (fridge). Once the squares or balls dry a bit you can put them on a grill like this and heat them. They become crispy on the outside and soft and sticky on the inside.
And from here you can decorate them any way you want.
You can put them into a sweet red bean paste soup.
Or sprinkle sweetened kinako (soy bean) powder on them.
Or wrap them in a sheet of seaweed and flavor with sweetened soy sauce.
Then there is always the old tried and true filling some mochi with sweet adzuki paste.
All the above are obviously desserts, but there is one non sweet dish that is served on New Year's day (and the days following) and it's called ozoni. There are all sorts of versions of this depending on where you live in Japan but it's basically a fish stock based soup with lots of veggies and chicken or some kind of meat with a chunk of grilled mochi dropped in.
And now if you really want to see something neat, here's a video of the mochi pounding. Watch until the end because they do a super fast thing where you won't believe how the one guy's hands don't get smushed. It's pretty cool.