Playing possum is an idiom meaning to feign death or "ignorance, indifference or inattention, with the intent to deceive; to dissemble; -- in allusion to the habit of the opossum, which feigns death when attacked or alarmed." (thanks to the possum entry of Websters 1913)

The etiology behind this idiom is a trick used by opossums to fool predators. Now, why would they do this? Because they are slow and can't outrun most predators. So the cute little critters put on a great show of being rotten meat. They lay on the their side, open their mouth slightly and draw the corners back while drooling. Their eyes stay open.

Research shows the heart rate and brain waves of an awake opossum and a possum feigning death are no different so the opinion that they have passed out in terror seems to be unfounded. Deer frozen in headlights however are not "playing possum". Headlights did not play an evolutionary role in deer's defensive strategies.

Most predators like their meat fresh. Some that aren't so particular will not eat it right away, but rather will cache the meat to eat later. Since the possum couldn't outrun the predator s/he hopes for the best and "plays possum"

Most of the above (minus deer coments and comments on possums' cuteness and hopes) was closely paraphrased from this URL:

"This is a very successful defense mechanism called thanotosis and many organisms (other than opossums) play dead, such as certain frogs, toads, and even insects. Frogs can remain so rigid and motionless that even respiratory movements is unobserved."

Another animal that feigns death is the click beetle. These little guys lay stiff, with antennae and legs all drawn in, looking like a desiccated shell of a bug, for long periods of time until they suddenly spring up into the air with a loud click, hopefully jumping out of range of the started predator. Ironically, a serendipitous run in with a click beetle saved me and the surrounding crowd from terminal boredom during an outdoor lecture about near death experiences. We passed the little creature from hand to hand 1 watching him vault loudly every few minutes and entertained ourselves while being regaled with tales of multiple lightning strikes and the revelations they produced in the speaker.

1) No click beetles were harmed during this entertainment.