Black and white moving character in a color video
You'd achieve the effect by masking one part of the image and applying a desaturation effect to it. The hard part is masking off one part of the image.

In software such as After Effects and Nuke you can mask off an area of the image with bezier splines, and the shape of the masks can be animated. Tracing and animating these masks is what is called rotoscoping (it was invented by animator Max Fleischer in 1915, where he hand traced frames of film projected onto paper). 

After Effects does come with an automatic rotoscoping tool called "Roto Brush" that works a bit like the magic wand in Photoshop, except that it changes shape over time. This can take a lot of the work out of rotoscoping, but it is far from infallible.