If the actual luminosity (the amount of electromagnetic power) from a star is truly changing, it is called an intrinsic variable. Intrinsic variables can be roughly divided into pulsating, eruptive, or cataclysmic variables, although some rare variable stars show more than one such phenomenon, like the R Corona Borealis stars.
A pulsating star, usually a giant star, alternately grows and shrinks in size because its outer layers move in and out due to the balance of the inward pull of gravity and the outward pressure of radiation coming from deeper layers of the star. As the star grows in size, its larger surface naturally emits more light. As it shrinks, it grows dimmer, and the cycle repeats periodically. One classic type of pulsating giant star is an RR Lyra variable star.
Eruptions on the surface of some stars are like super powerful versions of the solar flares that occur on our own Sun. Flares can cause large increases in a star's luminosity that occur unpredictably and might last for hours or days. A classic example of a flare star is UV Ceti. Other types of eruptive variables change brightness and blow out dusty winds that can strongly dim the light we see, as in T Tauri, Wolf Rayet, or R Coronae Borealis stars.
Some stars, like supernovae, literally explode, and are completely disrupted like a bomb. What's left is often just a dense core, like a neutron star or a black hole. Other explosive variability may happen when so much matter falls onto a dense, compact star (like a white dwarf or a neutron star) that thermonuclear fusion ignites on the surface, causing luminous explosions, but not destroying the star. Explosive variability may also be called cataclysmic.
Quasars are superluminous objects powered by active supermassive black holes at the centers of distant galaxies that display random changes in brightness.
Celestial transients are energetic bursts that can be millions or billions of times brighter than the sun. Light-emitting transients can be caused e.g., by the explosive deaths of massive stars, the detonations of white dwarf stars, or mergers of the densest types of stars, the neutron stars. Since transients may burst and rapidly fade, follow-up observations to study and understand them must be prompt - often on the same night, or within a day or two.