Which part of the eye controls how much light gets in?

The iris - the coloured ring - which opens and closes the pupil. The pupil itself is not a structure at all; it is a hole, and it looks black because almost no light that goes in comes back out.

Eye colour is not pigment in the way paint is. Brown eyes have plenty of melanin in the iris; blue eyes have very little, and are blue for the same reason the sky is, by scattering short wavelengths. There is no blue pigment in a blue eye, and the effect disappears if you light it from behind. Everyone with blue eyes appears to descend from a single mutation somewhere in the region of 6,000 to 10,000 years ago.

The iris also reacts to things other than light. It widens with fear, with certain drugs, and measurably with mental effort and with attraction - a response outside conscious control, which is why Renaissance women used belladonna to enlarge their pupils deliberately, and where the plant's name comes from. It is also unique enough between individuals to be used as a biometric, more distinctive than a fingerprint.