Mochi is a kind of gooey rice cake, which is eaten in many forms throughout the year. There are very soft sweet mochi cakes with red bean paste inside called Daifuku. We can also buy hard mochi in square shapes, called kirimochi. We heat kirimochi in a pan or grill over fire till it gets soft, then coat it with soy sauce, wrap it in nori and eat. Mochi is a big part of New Years in Japan. We buy a special kind of mochi before New Years called kagamimochi, which is one ball of hard mochi on top of a larger ball of mochi, making it look like a snowman. Kagamomochi is purchased sometime between Christmas and New Years, and it is kept uncooked until around the 7th of January, when it is broken into small pieces, cooked, and eaten. We call this ceremony kagamiwari or kagamibiraki , which means the breaking of the kagamimochi.
Mochi is popular during New Years because it has many connections to having good luck and good fortune. Another, more practical reason we eat mochi at New Years is that mochi, like many other kinds of New Years foods can keep from spoiling for a long period of time. In the old days stores would close for a week or more, so people had to make due with what they had in their houses for that period during the holiday. Another way we eat mochi during New Years is by putting pieces of kirimochi in a soup, which is called ozoni. Sometimes we can grill the mochi a bit before putting it in the soup, but basically it softens in the soup. This soup has many local forms. In the Kanto area (Tokyo) ozoni is a light soy sauce and bonito flake soup with square mochi. In the Kansai area (Osaka, Kyoto, Kobe) ozoni is a miso soup with round peices of mochi. And if you go to any area you're likely to find many other varieties of ozoni. So we have been eating lots of mochi in many shapes and colors these days.
Unfortunately, and somewhat tragi-comically, many older people in Japan die every New Years from eating mochi. It gets stuck in their throat and they don't have the strength to cough it up, so they end up choking on it. Already one person (that I know of) has died from mochi this year, on New Years day. The best way to save someone who is choking on mochi is to stick a vacuum cleaner hose in their throat. Apparently the Heimlich maneuver is ineffective in this case so remember to always keep a vacuum cleaner handy if you are eating mochi during New Years.