The Ursid meteor shower will be active between December 17 and 24 with its expected peak late December 21 into the early morning hours of December 22 (http://www.space.com/34973-ursid-meteor-shower-guide.html). Meteor showers are unpredictable in terms of the number and brightness of the meteors they produce, but if the weather permits, you might see some interesting meteors around midnight on the 21st up to dawn on the 22nd. It’s also worth looking for meteors on the nights preceding and following the expected peak since visible meteors can be seen during these times as well, although not necessarily as frequently as during the peak. The Ursid meteor shower gets its name from the fact that the meteors appear to generally radiate from the constellation Ursa Minor, although they can be seen in any part of the sky during the shower. The Ursids are associated with Comet 8P/Tuttle, which was discovered in 1790 and then re-discovered by Horace Tuttle in 1858.
If you want to capture images of the meteor shower, one option is to take a long duration exposure with a camera on a tripod aimed to the north. You will have to experiment to find the optimal exposure length as this depends upon the camera you have, but a sequence of a number of 30-second exposures will usually yield images that show plenty number of background stars with minimal trailing. With some patience, you might capture a bright meteor in one or more of the images.
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Mars and Neptune will appear less than ¼ degree apart from one another in the sky on December 31. This means that they will both easily fit in the same field of view in a backyard telescope, although Neptune will be quite a bit fainter than Mars. The pair will be easy to find since Mars will be a bright orange point of light in the west-southwest just after sunset, right above the planet Venus.
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The constellation Orion will be rising in the east all month, becoming visible in the east just after sunset. Orion is a spectacular winter constellation that contains a number of deep sky objects that are easy to find and view in a telescope. The crowd favorite is the Orion Nebula, just below the three-star belt of Orion, in the group of stars that make up his sword. This nebula is a region where new stars are forming and where young energetic stars, less than one million years old, illuminate the nebula to show one of the most interesting and most frequently imaged objects in the night sky. See http://earthsky.org/space/orion-nebula-jewel-in-orions-sword for more information.
Clear skies!
All images created by the author using Stellarium (0.11.3) software (http://www.stellarium.org/).