A variable star is a star with a brightness which varies from the perspective of a viewer on Earth. Some variable stars have variations so subtle that they can only be identified with advanced spectroscopy, while others vary enough that they can be seen with the naked eye or a basic telescope. Variable stars are of immense interest to astronomers, both professional and amateur, and more are being identified all the time.
Historically, people thought of the stars as fixed and unchanging. This theory was disproved when people began to observe novae, realizing that the contents of the heavens actually changed over time. A number of theories were developed to explain this, and astronomers began identifying variable stars. Initially, very few were developed, and they were thought to be relatively rare. With the advent of better observing equipment, astronomers realized that variable stars were actually quite numerous. Some examples of well-known variable stars include Betelgeuse, Beta Lyrae, Eta Aquila, and Polaris.
There are a number of types of variable stars. They are roughly broken into two major categories: intrinsic and extrinsic variable stars. Intrinsic variable stars have a brightness which varies because of internal processes occurring in the star. Stars actually vary quite a bit at the beginning and end of their lives as a result of a cascading series of processes which occur inside and around the star. Extrinsic variable stars vary because of factors outside the star.
In the case of an extrinsic variable star, the star's brightness may vary because of periodic occultation by another star, meaning that another star crosses its path and temporarily eclipses it. Stars can also vary in brightness as they rotate, if they have dark areas on their surface. Both of these types of variable stars have a very regular period, making it easy to track the waxing and waning of brightness.
Intrinsic variable stars are divided into cataclysmic, pulsating, eruptive, and x-ray variable stars. These stars are undergoing processes such as chemical changes which cause variations in brightness as the star swells and contracts, emits flares of energy, and so forth. Novae are a particularly notable example of a cataclysmic variable star, with a flare of brightness so intense that in the case of the largest novae, the star may light up the sky.
Amateur astronomers who are interested in variable stars can actually contribute to the body of scientific research on such stars. Amateur observation organizations collect data from backyard astronomers all over the world and provide that data to astronomers. Every now and then, an amateur astronomer even identifies a variable star or notes changes in a variable star before professional astronomers notice it; thanks to the huge number of celestial bodies, it's impossible for professional astronomers to keep track of everything at all times.