The calendar we currently use is the Gregorian Calendar. It was established by Pope Gregory in 1582 and followed by most of Europe under Roman Catholic rule. However, England (and by association, its American colonies), did not follow suit until September 1752 (England had its own church). This was the calendar that set the first day of the year at January 1.
The Gregorian Calendar was based on the Julian Calendar, which had been in use for centuries before; it, in turn, was a retrofitted version of the Roman Calendar, with the year adjusted to roughly correspond to the life of Jesus Christ. This calendar set the beginning of the year at March 21 (hence why February, at the time the last month of the year, was also the shortest and the one to which the leap day was applied). At the time, this was the same as the vernal equinox, the first day of spring. However, as the centuries passed, it became apparent that a certain "drift" was occurring-- the equinoxes and solstices that marked the seasons were occurring earlier and earlier. This meant that the calendar year was just slightly longer than the solar year (by, almost precisely, 3 days for each 400 years). The "leap year day" of February 29, which had been added to the calendar many centuries before because the calendar year was too short, had been observed every four years to fix this, and it almost worked. Pope Gregory proposed removing three leap years every 400 years, by removing the "leap year" designation from any year that was divisible by 100 (i.e. ended in "00") but not divisible by 400. The Pope retroactively applied this and allowed the official date to jump ahead 10 days, correcting the errors of the past centuries. (In England, where the change was applied later, 11 days had to be skipped, since that country had a February 29 in 1700 and the rest of Europe did not.)
By this time, the date of the "New Year" had changed to January 1 at
this time. It's possible that this was
chosen to bring it closer to Christmas, as the New Year traditionally
symbolizes birth and renewal.
Timekeeping and astronomy has become so precise in recent years that
international organizatons are now able to reduce the "leaps" necessary
to keep the calendar and solar years in phase to a mere second every
few years, virtually unnoticeable to most people.
The New Year is generally only recognized by nations dominated by
Catholic and Protestant Christians. Muslims follow the lunar calendar,
which is 11 days shorter than the solar calendar; as such, their
calendar drifts much faster (to the point where holidays that are in
the winter end up in the summer 16 years later). The Jews have a
lunar-solar hybrid calendar that adds an entire month to the calendar
each leap year. Russian Orthodox Christians still use the Julian
calendar, never adopting the adjustments, and are about two weeks
behind the Gregorian calendar.
The tradition of dropping a ball on New Year's Eve dates to the use
of time balls to
synchronize timepieces in the 1800s. At observatories,
a ball would be placed at the top of a tower and slowly lowered so that
it reached the bottom of the tower in a specified period of time
(usually one minute).
New York City, in the early 1900s, revived this tradition as a
symbol of the new year, by dropping the Times Square Ball from the
top of Times Square at 11:59 Eastern Time and having it land at
midnight to significant fanfare. It has been celebrated every year
since in Times Square, with the exception of a few years during World
War II. The event grows bigger each year, with over a million visitors
in 2007-08. Other cities are known to celebrate ball drops as well: in
Atlanta, they drop a replica of a peach each year, known as the Peach Drop; it is the
largest event of its kind in the Southeastern United States and draws a
crowd of tens of thousands. Buffalo boasts a similarly sized
celebration known as the Buffalo
Ball Drop, which has been happening since 1988-89 and
currently draws a crowd of approximately 40,000. Both are carried on
local television. More bizarre, perhaps, are the 100 or more local
imitators of these ball drops. Wikipedia has a
Other celebrations, usually consisting of only fireworks, exist in
Seattle, Toronto and Las Vegas (Fox News covers Las Vegas events), and
the First Night organization holds celebrations across the country.