There's a lot of interest going on in the sky this month. The sun's the obvious one, to most. The winter solstice is this month. Where I am, it'll be the shortest day of the year, with just under 9 hours from sunrise to sunset. At solar noon, when the sun is as high in the sky as it will get that day, it will be at the lowest peak point for the year. A few days after the solstice solar noon will slowly start to climb higher and higher in the sky.
The shortest days of the year also bring the longest nights. This month's full moon will be on the 28th. The sun and the moon do an interesting little dance - while the sun is staying close to the horizon during the day, this full moon will be higher in the sky than any other full moon in the year.
There is of course more of interest beyond the sun and moon, for those who are willing to look a little more carefully. In the early morning, a few hours before sunrise, Mercury and Venus can be seen in the east. Saturn can also be seen in the morning sky. In December Jupiter reaches it's yearly peak, and can be seen for a good portion of the night.
On the 13th and 14th of December the Geminid meteor shower will be at it's peak. Last year's show was obscured by the full moon's light, but this year the new moon is on the 13th, which will make for a clear view.
The Geminid meteor shower is named after the constellation Gemini, which will be easy to spot in the sky. Taurus will be nearby. Both constellations will be near Orion, which is a fairly clear and easy to spot/learn constellation.
For many it's a cold time of year, especially at night, so if you head out to look at the sky remember to bundle up (usually more than you'll think is necessary), and if you're going to be out there for a while - say, watching the meteor shower - bringing something like a thermos of hot cocoa, coffee, or tea can be a nice way to help keep warm.
(If you're interested in seeing some specifics for your local area, head on over to Time and Date - under "Sun & Moon" you can see sun and moon rise, sun and moon set, length of day, solar noon, exact moon phases and solstice timing, and so on.)