What is the general relationship between temperature and star brightness?

Stars in their Main Sequence stage have generally proportional temperature and color. The color-temperature spectrum of a star ranges from red (2000-3000 Kelvins) to blue (25,000+ Kelvins). Red Giants have a relatively high luminosity and low temperatures. White dwarfs have relatively low luminosity and high temperatures. Main Sequence stars are proportional temperature/color therefore they can vary from relatively high luminosity and temperature to relatively low luminosity and temperature.

Absolute Magnitude is the star's genuine brightness. It's apparent magnitude is it's brightness from earth. A star can only be accurately classified once data on it's absolute magnitude is acquired.