January 1 is used to remember the old and ring in the new year.  It is generally marked by the Gregorian Calendar and celebrated with parties and fireworks.  It is often a time of reflection and resolutions to stop drinking, smoking or lose weight.  The Rose Bowl football game is played that day as are many other college bowl games.  Parades such as the Tournament of Roses or Mummer's Day Parade celebrate the day.

New Year's Day is celebrated in many countries; Israel, Canada, United Kingdom, Switzerland, Sweden, Russia, Australia, Wales, Cuba, China and the Phillipines and many others all holding their own celebrations.  We all know the image of Father Time (or the "Old Year") wearing a sash across his chest with the previous year printed on it passing on his duties to the Baby New Year (or the "New Year"), an infant wearing a sash with the new year printed on it. Hospitals give prizes to babies born on New Year's Day.

New Year's Eve is a separate observance from the observance of New Year's Day. In modern Western practice, New Year's Eve is celebrated with parties and social gatherings spanning the transition of the year at midnight.  It is often celebrated with fireworks.  How many of remember Dick Clark counting down the final minutes of the year?  At the stroke of midnight, a giant ball is dropped down in Times Square.  We toot horns and sing or dance to bands (who can forget Guy Lombardo) to "Auld Lang Syne."

http://en.wikipedia.org/wiki/New_Year%27s_Eve