In many older calendars, the start of a new year was actually celebrated in spring, tying it into the rythmns of the earth. If you think about it, 1 January is purely arbitrary and not really connected to anything, but it was the Romans who gave us that date.

The Romans at first observed New Year in spring. The Roman calendar got out of whack, both because time was being lost, and because of political tampering over time. In 153 BC, the Roman Senate approved a re-ordering of the calendar, fixing 1 January as the New Year. That date was reinforced with the Julian calendar passed in 46 BC. The Church was remained opposed to any New Year's celebrations until around 1600 AD.

Most North Americans, in both Canada and in America, watch the celebrations in Times Square, New York on television. The ball-dropping started in 1908. In Calfornia, the Tournament of Roses Parade was first held in 1886. New Year used to be a time of very heavy drinking, before tough drunk driving laws came into effect.

Some foods are seen as lucky to eat at New Year. The foods, though, vary by culture:
Austria: Pork
Brazil: Lentils
Denmark: boiled cod
Dutch: doughnuts (Olie Bollen)
Southern US: black-eyed peas
Spain, Cuba, Mexico, Venezuela: 12 grapes at midnight, one for each of the upcoming months
Pennsylvania: Sauerkraut

In Japan, New Year is called either "Shogatsu" or "Genjitsu", and celebrated on 1 January, as it is in the West, though the big celebrations are held on New Year's Day, not the evening before. In fact, many people go to bed early on the 31st December, so they can wake before sunrise on 1 January.
Traditional clothing is worn on New Year's Morning. Buddist Temples toll their bells 108 times. New Year's food is called "osechi-ryori." You can make it yourself, or place orders with restaurants or stores to have it all delivered on New Year's morning. Soba noodles, ozoni and mochi are served for luck. Every year, some people die in Japan from having the mochi (sticky rice cakes) choke them to death.

Before New Year, houses in Japan must be completely cleaned, and people send New Year's greeting cards (though many now send E-Cards instead.) The celebrations go on until 3 January, with lots of visiting.

Language Notes
Most people commonly say "Happy New Year's!" when in fact what they mean is "Happy New Year!". There is no reason for "Year's" to be possessive.