A classic professional wrestling maneuver. It is a submission move, attempting to make the victim pass out or give up to lose the match. It is performed by having the bearhugger wrap his arms around the mid to upper torso of the victim, under the arms, and squeezing tightly. Optimally, this is accompanied by lifting the victim off his (or her) feet. Optinally, the bearhugger can shake the victim around a little, allowing the victim to flop around lifelessly in order to build drama in the match.
The rationale of this move is that it squeezes the breath out of the lungs of the victim, and a bearhug could probably do so, if the bearhugger continually tightened his (or her) grip with each of the victim's exhalation.
The great weakness of this move, both in professional wrestling and real life, is the fact that the victim's arms and legs are free to do whatever they want - such as sticking the fingers into the bearhugger's eye, nose, ear, etc., or kicking the bearhugger's nads.
This move was prevalent before the "wrestling renaissance" of the 1990's. Before then, there were a lot more non-physically fit wrestlers. The bearhug allowed both wrestlers to make faces to each other and the crowd while catching their breath. With the advent of physical conditioning and high-paced, short matches, this resting maneuver was used less and less. This is good from a spectator standpoint, because it is boring to watch - the only interest being from the look of agony on the victim's face.
I hadn't seen a bearhug in a few years, then suddenly, it was used by Brock Lesnar to put
"Hollywood" Hulk Hogan out of commission... This was a bit odd, as Brock could have just picked up Hogan and tossed him around instead. Then again, the bearhug allowed the audience to see the look of agony on Hogan's face, and allowed Hogan to sell his lower back injury.