What part of the microscope controls how much light is allowed to pass through

The piece that allows you to adjust the amount of light that's coming through the microscope is called the adjustable diaphragm.

Just to elaborate further, the adjustable diaphragm was invented by a famous scientist from the mid 1600's ( Julian Calendar )named Robert Hooke. He was accomplished in many fields and invented new instruments besides the diaphragm, such as the balance wheel and universal joint. He formulated Hooke's Law, which stated the following: that the extension of a spring is directly proportional to the mass of an attached load. Hooke was also a natural philosopher and was a early proponent of the theory of biological evolution.