2013 stands to be one of the best years for comet viewing. First up: C/2011 L4 Pan-Starrs.

This comet has already been visible in the southern hemisphere (comment if you’ve seen it!), and is expected to TODAY (March 8) be more easily visible to us in the Northern Hemisphere. My husband has our telescope and binoculars ready-to-go, waiting for a clear night. The best viewing should be in a few days, the 12th and 13th of March.

C/2011 L4 Pan-Starrs should be visible for the rest of the month, down and to the left from the moon just after sunset.
Comets are made of collections of rocks, dust, and ice. Most start as part of the Oort Cloud or Kuiper Belt and are a few km to 10s of km across. Sometimes, these masses of stuff begin to orbit the sun and as that happens, the comet begins to sublimate (the “stuff” goes from solid to gas) and falls apart as a result of solar radiation and (solar) wind, creating a tail. The tail of a comet always points away from the sun and does not indicate its direction of travel.
For more details, see these articles
Sky and Telescope: “Comet PanSTARRS Still Punking Out”
by BBC News: “Bright comet ‘lighting sky’ as it flies by Earth”
the New Zealand Herald: “Rare comet to be visible in NZ skies”