All stars are variable, that is, their energy output is not constant over time. Even our Sun, so nearby we have been able to monitor it in extreme detail, has a tiny but measurable 11 year energy variabilty associated with its sunspot cycle. This is probably superimposed on other cycles, both regular and random, which operate over much longer periods of time. In addition, there are random short-term events, such as solar flares, which temporarily alter our star’s energy output, both in total, and at specific isolated wavelengths of light. But every star, and our sun is no exception, go through periods of pronounced temporary variability during certain periods of their evolution as they adjust to changing internal conditions. Remember, “short”, “quick”, “long” and “termporary” in astronomical terms are relative terms. We must never forget that over a human lifetime, and even over millions of years of history, these changes are very slow. The universe is an active, evolving place, but it all occurs at time scales difficult for us to comprehend. At any one time, we are only examining one frame of a very, very, long movie.

Having said that, our sun, like the vast majority of other stars, seems to remain remarkably constant in its energy output for very long periods of time. After all, life has existed on Earth for close to four billion uninterrupted years.

Why are stars so stable for so long? A star is in a constant state of balance. It’s immense gravity tends to want to collapse it to a point, but thermonuclear reactions and the property of a gas that it tends to heat up and increase pressure when compressed, push back: the Gas Laws. Most stars remain in thermal and hydrostatic equilibrium, and change very slowly as a result. Internal conditions change when the concentration of thermonuclear fuels drops as they are consumed, or increases as gradually changing conditions make different thermonuclear reactions involving different elements possible. These reactions depend not only on the star’s internal chemistry, but its temperature, pressure, density and changes in radiative transport due to the ionization states of different atoms in the plasma at different depths. It is an exquisitely complex process.

Stellar variability seems to be most common in highly evolved stars where different nucleons are undergoing fusion at different levels and the energy-producing layers of the star are arranged like the layers of an onion. A gradual change in one leads to a change in all the others as the star adjusts, and the star may find itself oscillating as first one, then another, layer increases or decreases its fusion rate. Sometimes a change in one leads to a counteracting change in another, and the star pulsates, with periods varying from years to hours. These states of dynamic equilibrium are usually temporary, but even a pulsating star can be stable–unless some sudden change of state becomes runaway, feeding on itself; the variation hits a positive feedback loop. Stars can suddenly change, too, or even explode.

But in general, the process is mostly gradual and stability is preserved. If the star cools, it shrinks under gravity and heats back up, which provokes more fusion. If it gets hotter, the star will expand then cool, lowering the fusion rate, causing it to shrink back. The process is remarkably sensitive and extremely successful at keeping the star essentially constant over long periods of time. When stars run out of fuel, they shrink until only back pressure can hold up the weight of the outer layers–or until some catastrophic internal change of state suddenly intervenes and the star goes out of control.

Most of what we know about stellar structure and evolution comes from our study of variable stars, and although we are too short-lived to follow any one star throughout its life cycle, there are so many of them at different stages of their evolution that we have been able to stitch together an (admittedly) vague and incomplete picture of what is going on inside stars over billions of years. It is one of the triumphs of the human intellect, and most of it has occurred within my lifetime.

So when you consider many different stars of highly varied initial mass and chemical composition, of different ages, you can expect to find many of them moving into or out of periods of variability. Again, I rely on the analogy of a stroll through the rain forest, and by studying the populations and proportions of different plants, we can piece together a narrative of the forest going from a burnt-off waste to a stable climax community. Most stars are stable, but a few are variable, and they seem to fall into a large but finite number of categories. A few seem really bizarre and unique: There are four basic types: regular, irregular, eruptive and catastrophic variables, and they can be identified by their light curves, a graph of their brightness variation over time.

Regular variables are in a state of dynamic (as opposed to static) equilibrium. They get bright, and then they get dim, their light curves similar to a simple harmonic oscillator, a sine wave, with remarkably stable amplitudes and periods. A star can vary its light output by an order of magnitude or more, over periods ranging from years to days, or even hours. There is much variation in that broad category, and regular variables are further classified by the distinctive shapes of their light curves. As the light varies, so does their color, and we can also measure the Doppler effect in their spectra and determine how fast and how deeply the star breathes. In some stars, several different light curves appear to be superimposed on each other, as if separate, uncoupled oscillations are going on simultaneously.

Some regular variables. like the Cepheids, are so predictable and their periods so correlated with their brightness that they can be used as standard candles to measure distances. The first evidence of external galaxies was the discovery by Hubble that some very faint Cepheids barely visible in the Andromeda Galaxy were shown to have oscillatory periods identical to those of very bright Cepheids in our own galaxy–stars whose true distances and brightnesses were known. In a flash, we were able to determine the distance to that object, and prove Andromeda was not a small local cloud near the Milky Way, but another island universe altogether, and that all those mysterious “spiral nebulae” in time exposure photographs were actually other Milky Ways, stretching off to infinity. I have often wondered what the psychological impact of that realization must have been to the first human being who grasped and fully understood it. Just thinking about that moment of blinding comprehension brings tears to my eyes, and a shiver down my spine.

Irregular variables change, but there is little or no pattern to them, or, a repeatability can be detected, but it is more stochastic in nature. Needless to say, it is much harder to come up with the internal physics of these objects. Eruptive variables are those that remain generally stable but occasionally and unexpectedly experience some outburst. Solar flares are an example of the most gentle of these episodes.

Catastrophic variables are the novae and supernovae. The former are a massive and abrupt change in brightness, an explosion, after which the star settles down relatively unchanged for a long period of time. Some, perhaps all, novae occur periodically during certain stages of some stars’ evolution. There are several different types of novae and supernovae.

A supernova blows the star apart, it is evidence of a star’s internal feedback mechanisms failing completely. Most of the star’s material is blasted off in an explosion that begins in seconds, lasts for months, and can easily outshine the entire galaxy it occurs in. All that survives is the remains of the star’s core, a white dwarf, a neutron star, or a black hole. They are the most energetic events since the Big Bang and can be detected to the very edge of the known universe.

There are dozens, perhaps hundreds of different types of variables, including many one-of-kind types, transitional types, combinations of two or more types, etc. Even variability is variable.
This essay has just scratched the surface. If you are interested, I highly recommend the Wikipedia article. It is excellent.

https://en.wikipedia.org/wiki/Variable_star