The function of lasers is to generate artificial stars in the sky. By firing the laser into the atmosphere and constantly monitoring how the laser light is distorted, the telescope mirror can be deformed to correct the observations in real time.
A supernova is the explosion of a star. They are usually very massive stars that at the end of their lives end up expelling all the material that was inside them by means of a shock wave and this allows us to see the different layers that made up the star. There are different types of supernova that explode in different ways. Type Ia supernovae are the ones that explode thermonuclearly. They are basically binary systems, two stars orbiting together, in which one of them, when it is very old, absorbs material from its companion. Another type of supernovae are called type II supernovae, which are very massive stars that during their lifetime maintain a balance of forces between the energy they produce, which is centrifugal and gravity, which is a force towards the interior of the star.
Planets orbiting other stars are called exoplanets. Exoplanets are very difficult to see directly with telescopes. They are hidden by the bright glow of the stars around which they orbit. Therefore, astronomers use other ways to detect and study these distant planets. Kepler detected exoplanets using the transit method. Astronomers can observe how the brightness of a star changes during a transit. This can help them work out the size of the planet.