Etymology
English play + possum (from opossum), from the behaviour of the Virginia opossum, feigning death when threatened. First documented 1822.

(idiomatic) to feign death; to remain quiet and still to escape attention or remain undetected; to lay low.
Thinking fast, we played possum hoping the bear wouldn't bother us.
The soldier played possum, fooling the sniper.
To keep the focus away from his client, the lawyer basically played possum during the entire complex trial, and his tactic paid off with an acquittal.

(idiomatic) to feign sleep, illness, etc.
When we used to get home late at night, I would play possum so my daddy would carry me inside and put me in bed.

(idiomatic, dated) to dissemble or to feign ignorance; to disguise or conceal something in order to deceive.

A possum will often act dead when faced with a situation it doesn't want to deal with.

Playing possum, when someone comes to your house that you don't want to talk to, so you act like your not home.

Joe "Hey greg said he dropped over your place earlier, weren't you home?"
Darren "I was home, just didn't want to talk to him so I was playing possum!"