The exchanging of nengajou (New Years cards) started in 1873 when postcards were first introduced to Japan. It caught on big. The average family sends over a hundred nengajou to their relatives, friends and colleagues. Virtually all businesses mail new years cards to their customers. To get an idea as to how many cards are sent, nengajou accounts for almost 20% of all annual postal revenues. Approximately 35 billion cards are sent every year.
Postal workers in Japan do not get New Years day off, since it is the busiest delivery day of the year, they do get December 31 and January 2nd off though. Post offices in Japan will hold nengajou so as to deliver them on January first. To ensure that post cards are delivered on January first the post office in Japan has taken steps like placing special temporary mailboxes adjacent to permanent ones at post offices for nengajou. Every card put into the temporary boxes between December 15 and 25 gets a special postmark and is delivered promptly on New Years Day.
Cards can be made or purchased at many places but government postcards are available only at post offices for 50 yen each. These government postcards are blank on one side, where you would write your greeting, and have a lottery number and a new year postage stamp printed on the address side. The New Years lottery cards were first sold in 1949. Demand for lottery cards exceeds supply, they usually sell out in the second week of December (they print around 4 billion of these lottery New Year cards every year). The lottery is drawn on TV, and the winning numbers are published in newspapers on January 15. If you win you can get your prize at the post office. Prizes vary, but some typical prizes are commemorative postage stamps, televisions, and of course cash.
Popular pictures on cards are illustrations of the animal for the coming year under the Chinese zodiac. After that are New Years motifs such as kadomatsu (decorations made of pine branches), kites, plum flowers, and the sun rising against Mount Fuji.
A bit of etiquette: If you get a card from someone you did not send a card to, it is customary to send a card on January 6, formally asking them to take care in the cold weather. If there has been a death in the family, it is customary to send a mochu (bereavement card) to inform people they shouldn't send you nengajo.
It is customary to write Akemashite omedeto gozaimasu (Happy New Year) on the cards and what you've been up to lately. For variation you can also write kinga shinnen the more formal version of Happy New Year or shinshun no oyorokobi o moshiagemasu meaning: I would like to wish you a pleasant spring.
-The picture above is taken from Azumanga Daioh volume 2 and displays a nengajou (though it is a custom made one not a purchased one with the lottery on it). Teachers giving New Years cards and receiving them from students is common.