Kagami Biraki (鏡開き) is a Japanese traditional ceremony which literally translates to "Opening the Mirror" (from an abstinence) or, also, "Breaking of the Mochi." It traditionally falls on January 11 (odd numbers are associated with being good luck in Japan) but, in practice outside of Japan, generally occurs around that date. It is generally the first important event of the year after New Years Day. It refers to the opening of a Kagami mochi, or to the opening of a cask of Sake at a party or ceremony.
The fourth Tokugawa Shogun was the first one to hold this ceremony 300 years ago. Before going to war he gathered his daimyo in his castle to break open a sake cask. The battle was successful, so from there on a new ceremony was born.
The ceremony nowadays is also performed at weddings, sporting events, starting a new company, etc.
In Japan, mochi was traditionally made at home, but most families today buy it ready-made. Over the holidays, a pair of round mochi (kagami mochi) the size of small plates -- one a little larger than the other -- is stacked on a stand and placed in a household Shinto altar or tokonoma as an offering to the deities that visit on New Year's. The ornamental mochi is removed on January 11 and broken into smaller pieces before being eaten.
By this time, the kagami mochi is usually quite brittle, and cracks appear on the surface. The mochi is not cut with a knife, since cutting has negative connotations (cutting off ties) and is instead broken with one's hands or a hammer.