What's the best way to insulate a window?

You could, but first, check around the window frame for leaks. Caulk any you find, outside and inside. Then get some foam insulation strips and put them in the space between the bottom of the window and the top. You don't say if these are wooden double-hung windows--if they are, you need to also insulate at the top where the top sliding window fits into the frame, and the space between the two sliding parts of the window--the foam strips fit nicely into those spaces. If there are still drafts and leaks, the window probably has shrunk in the frame--you will need to use a sheet of plastic over it on either the outside of the house, or the inside in the room. Or both.

Is the heat on in that room? Perhaps the room wasn't being used and someone closed off a damper?

Check around the window for air infiltration. You can use a smoke stick or a stick of incense to look for airflow. Caulking and/or weatherstripping will help eliminate that.

A storm window will help, as will those plastic window films.