喪中 (mochuu) is a Japanese word that translates as "in mourning." When a family member of a first- or second-degree relation to you (first-degree = parents, children, spouse; second-degree = grandparents, siblings) passes away, in Japan it is traditional to honor the deceased with a one-year mourning period. (Case by case, families may choose to identify themselves as 喪中 for deaths of more distant relatives, like the spouses of siblings or children, as well.)
Over the New Year's celebrations of that year, the mourner sends out no greeting cards. Instead, at the beginning of the winter season they send cards informing their friends and coworkers of the loss.
Now, slip ups happen. Sometimes you send a card to someone you haven't been in touch with for a while, and family tragedies can occur any time, even right before the new year. If you accidentally send a card to someone who is in mourning, the proper recourse for them is to reply to you after the 松明け (matsuake), a old Japanese calendar event that marks the ending of the New Year's season. (Matsuake, by the way, is on January 7th in Eastern Japan and January 15th in Western Japan.)
Their reply to you will be as a normal postcard, and they'll probably inform you of the loss and apologize for not being able to send you a New Year's card that year. Since this is the first time you're hearing of the loss, it's polite of you to at this point send one more postcard to them, avoiding any congratulatory language and telling of your condolences.