"Under the hood " also refers to training to fly aircraft in instrument conditions. In order to receive a Private Pilot's license in the U.S. you must fly at least three hours in such training before your final check ride. It's called this because it does what it says on the tin - while training in this manner, two pilots are always in the airplane. The one training and a 'safety pilot' (usually a certified flight instructor but private pilots can serve as each others' safety pilots for refresher work). The pilot in training puts on a device to prevent them from seeing clearly out of the airplane. There are a variety of such aids available - the classic 'hood' is a plastic visor which restricts the pilot's vision to the instrument panel of the airplane and anything inside. Also used are more 'goggle'-like versions - opaque binocular shapes which restrict the pilot's vision to a narrow cone straight ahead. Finally, since IMC doesn't mean no light, there are what is known as foggles - safety-goggle like devices which have their upper parts frosted over, so that the wearer can only see clearly in front of them and below midline.
The reason for this training is to ensure that if a VFR pilot accidentally flies into a cloud, they will be able to control the aircraft without outside references. The recommended procedure, should you do this, is to immediately make a 180 degree standard rate turn and return to clear air, and either avoid the clouds by going around them or finding somewhere to land and wait them out if necessary. Generally, flight training will include turns, climbs, descents, level flight, and perhaps rudimentary radio navigation while under the hood, to mimic maneuvers and activities you may need to perform to safely exit the cloud or to comply with ATC instructions when you declare your distress. In addition, the instructor will place the airplane in 'unusual attitudes' and then have the student recover to level flight, to simulate becoming disoriented upon entering clouds.
If the pilot trains for an instrument flight rating, then the hood is used as well - this time, however, the pilot is expected to navigate and if necessary land the aircraft rather than simply demonstrate control.