The winter solstice is, technically, not the shortest day of the year, but the brief instant at which the Earth's axial tilt is farthest away from the sun; this occurred on Dec. 21, at 17:47 UTC. More colloquially, it is the day on which this instant occurs, which is commonly known as the first day of winter. So, today is the 11th day of Winter. The solstice can occur at varying times; in 2006, it happened 22 minutes after midnight UTC, according to the USNO. The solstice being an instant rather than a day can be a source of some confusion, as when pagans celebrate the solstice on the wrong day, showing up at Stonehenge a day early (since it seems Stonehenge observes the first sunrise after the moment of the solstice).
Measuring this precise moment is difficult for non-astronomers, but ancient peoples built many structures aligned with the changing position of the rising or setting sun throughout the year. The summer solstice is easier to determine using their methods than the winter, because around the winter solstice, the "sun stands still". More modern instruments can use this phenomena, the analemma, in an analemma calendar.
This New Year's Eve will also see the second full moon in the month of December, a "blue moon" (1, 2, 3, 4). The next winter solstice, on December 21, 2010, will feature a total lunar eclipse visible in all stages throughout North America.
I reckon everyone will celebrate the end of the Aughties tonight, since 2010 begins in a few hours. Of course, the 10th year is the final year in a set of ten years; 2010 is the last year of the first decade of the 21st century, and the second decade begins in 2011. We begin counting with the number 1, not the number zero. But the common parlance will have tonight as the end of the Aughties, and (not being a fan of the decade) I'd say we should move on to a new one sooner than later.
Happy New Year!