The derailleur hanger / mech hanger is the part of the dropout that the rear derailleur attaches to.
Most non-steel framed bikes have a separate removable derailleur hanger, generally made from aluminium which is bolted to the dropout. These are intended as a cheap replaceable part so that in the event of an accident or mechanical problem that could damage the derailleur or frame, the derailleur hanger breaks or deforms instead. Sometimes a shear bolt which is designed as a weak point will also be used. There are dozens of different derailleur hangers available and one model is seldom interchangeable with another.
In general, steel-framed bikes do not have a removable derailleur hanger, because a steel dropout and hanger is stronger and therefore less likely to be damaged, and also it is more malleable and less likely to work harden during deformation and generally can be bent back into shape without breaking.
It is important for proper indexed shifting for the derailleur hanger to be properly aligned. The rear derailleur bolt hole must be close to parallel with the rear axle. If it is out of alignment, the rear derailleur will not move far enough, with respect to the rear sprockets, with each click. A special tool exists to measure and correct misaligned hangers, referred to as A DAG - Derailluer Alignment Gauge.