Kagami mochi 「鏡餅」 is a special traditional Japanese decoration for the New Year, usually displayed inside the house in the kamidana, for Toshigami, the god of the new year, to bring good luck and prosperity in the new year.
Kagami mochi is made from two rice cakes (mochi) of different sizes, the smaller placed over the larger one, and a daidai, a Japanese type of bitter orange placed on top. In some versions, the mochi are placed on a konbu sheet, a symbol of pleasure and joy.
As I wrote when I described the shimekazari New Year decoration, a kanji writing for daidai 「代々」 can mean “generation to generation”, so the small orange symbolizes the continuity of the generations and long life, while the mochi symbolizes the past year and the year to come. So, kagami mochi symbolizes the continuity of the family over the years.
The name has also an interesting explanation: the shape of the two mochi is similar to the shape of the copper round mirrors (kagami in Japanese) used during the Muromachi period – you can see several photos at the Kyoto National Museum.
The kagami mochi is kept until the beginning of January when, on 11th, (or on the second Saturday or Sunday from January) a Shinto ritual named Kagami Biraki (the opening of the mirror) takes place, the first important ritual after the New Year. Then, the kagami mochi is broken, with the hand or with a hammer, into edible-size pieces and it is cooked (e.g. shiruko). A knife is never used, because that would mean cutting the family ties.