The celebration of the new year is the oldest of all holidays. It was first observed in ancient Babylon about 4000 years ago. In the years around 2000 BC, the Babylonian New Year began with the first New Moon (actually the first visible cresent) after the Vernal Equinox (first day of spring).
Although in the first centuries AD the Romans continued celebrating the new year, the early Catholic Church condemned the festivities as paganism. But as Christianity became more widespread, the early church began having its own religious observances concurrently with many of the pagan celebrations, and New Year's Day was no different. New Years is still observed as the Feast of Christ's Circumcision by some denominations.
One of the most venerable New Years traditions is the champaign toast at midnight to ring in the new year. Toasting can be traced back to the ancient Romans and Greeks who would pour wine, to be shared among those attending a religious function, from a common pitcher. The host would drink first, to assure his guests that the wine was not poisoned. Poisoning the wine was a fairly common practice in ancient times, designed to do away with one's enemies. In those days the wine was not as refined as it is today so a square of burned bread (toast) would be floated in the wine bowl and then eaten by the last person to drink. The bread was put there to absorb the extra acidity of the wine in order to make it more palatable. Eventually, the act of drinking in unison came to be called a toast, from the act of "toasting" or putting toast into the wine.
The song, "Auld Lang Syne," playing in the background, is sung at the stroke of midnight in almost every English-speaking country in the world to bring in the new year. At least partially written by Robert Burns in the 1700's, it was first published in 1796 after Burns' death. Early variations of the song were sung prior to 1700 and inspired Burns to produce the modern rendition. An old Scottish tune, "Auld Lang Syne" literally means "old long ago," or simply, "the good old days." The lyrics can be found here.