The MR in MR16 stands for multifaceted reflector, which is the type glass housing for these lamps. They are typically halogen, a type of incandescent lamp, but LEDs and Fluorescents can be MR lamps, also. The reflector controls the direction and spread of light cast from the lamp, which can vary from very focused to wide. The 16 represents the size of the diameter in eighths of an inch, so an MR16 = 2 inches.
Designers use MR16s because they are typically smaller with better beam control and have a whiter light than other incandescents, but they can also be quite hot. An LED MR16 would vary in color but not be hot.
They are usually a type of low-voltage lighting, which means they require a transformer to step the line voltage down from 120V to 12V.