An emergency descent in an aircraft is pretty much just what it sounds like - a maneuver intended to reach a lower altitude as quickly as it is safe to do so. There are any number of reasons this might be necessary- if the aircraft has a system failure and needs to land as soon as possible, for instance. Or in pressurized cabin aircraft or aircraft with supplemental oxygen, a loss of cabin pressure or a failure of the oxygen system might require an emergency descent to place the aircraft low enough for the crew to remain conscious without the oxygen systems. It is possible ATC may instruct a pilot to perform an emergency descent in an emergency situation; although it is devoutly to be hoped this is never necessary, if so instructed, obey your controller without hesitation because something has gone wrong.
Whatever the reason, this maneuver will be described, and the procedure for it detailed, in the aircraft's Operating Handbook. The procedure is set by the manufacturer or by the aircraft operator. In general, however, it will involve retarding engine power to minimum and then pitching the aircraft forward until the airspeed reaches the designated Vno or maximum normal operating speed. The pilot will then use pitch to maintain this airspeed, increasing pitch to slow and decreasing pitch to accelerate. If the aircraft has designated procedures for minimum power operation or descent (for example, the use of carb heat or ensuring that the mixture is on full rich) those will likely also be specified for emergency descents.
During the descent, if flying VFR, the pilot should make a series of shallow turns left and right to check for any nearby traffic.