How do the mirrors in lasers resist destruction by the laser? Are lasers limited by the quality of the mirrors they use? How are the mirrors used in a laser not destroyed or ignited in the process? Are mirrors just really resistant to light? Is the mirror user in a laser same as the silicon dioxide in my windows? 
 A: Lasers use first surface mirrors. The "glass" substrate is not exposed to the laser light (except, for the exit mirror, which transmits only a small fraction of the light from the resonant cavity.)
The actual reflective surface is a thin film coating that is extremely efficient at the laser's wavelength (again, except for the exit mirror, which transmits a controlled small fraction...).
The "glass" used in the exit mirror and, in any lenses, often is a specialized material (in some cases, it may be a single crystal of the material) that is extremely transparent to the laser wavelength.

See also: https://en.wikipedia.org/wiki/Laser_damage_threshold
A: I initially wrote this as a comment but I'll write it as an answer as well for the sake of it. 
Mirrors used in laser cavities have extremely high reflectivities and thus a very small portion of the total cavity energy is absorbed in the mirrors as a large portion of what is not reflected is instead transmitted as the output. There are other elements in the cavity that might need active cooling, such as the gain medium etc. 
