Variable stars are Stars that change in magnitude that can be a minor change to something that is very different. The reason for the change can be for instance, a change to the internal forces within itself or an external source. There are different types of variable stars as described below. Whilst each variable star below can be broken into subclasses again, I am only doing a high level explanation of what each group is. the Sun is a normal, non-pulsating star, it is called a main sequence star, one where the forces that cause stars to contract and expand are in equilibrium so in other words, the forces cancel each out one. Some very well know stars are variable stars such as Betelgeuse and Antares.
Pulsating Variables are the stars that expand and contract according to what is going on inside the star. The stars can expand and contract as a sphere or they change their shape to a non-spherical shape. Those that expand as a sphere are Radial and those that can bulge or expand and contract in just parts of the star are known as Non-Radial stars. The type of pulsating variable is down to how long they take to pulsate.
Cepheid Variables are named after the star Delta Cephi, also known Alrediph in the constellation of Cepheus by John Goodricke, British Astronomer in 1874. These stars will tend to pulsate between around five to ten days. Class I Cepheids tend to be more luminous their Class II W Virginis Cepheids. Type II pulsate for longer, usually about 12-20 days. These second type stars tend to be older and have less metallicity in them.
These are short pulsating stars where they can expand and retract in a period as little as an hour to a day. These variables are normally found in star clusters and have been used to calculate distances to these star clusters. They tend to be older and less luminous than the Cepheid Variables. They are named after the star RR Lyrae in the constellation of Lyra.
Name after the star RV Tauri in the constellation of Taurus, these pulsating stars are yellow supergiants that a pulsate can take anywhere between 20 and 100 days.
These are typically red giants or supergiants that can take years to pulsate from minimum to maximum. These type of stars are referred to as Mira or Ceti stars. Mira in the constellation of Cetus itself is a star that a period of 331 days and which what this group gets its name from.
Whilst all the others are regular pulsating stars, there are some stars which do not pulse regularly and are known as Irregular/Semi-Regular stars. The stars Antares and Betelgeuse are both examples of these types of stars.
These Variable stars brighten sudden due to the inner operations of the star. They can be the result of the death of a star when the star explodes and turns into a neutron star. nebula or a black hole.
Supernova is an one time only variability of a star that what happens when a star has come to the end of its life and then explodes because the outward pressure over-rules the pressure to keep itself in. The Sun will one day supernova but that's not for five billion years yet at the current estimates.
A nova is typically a white dwarf that is orbiting round a larger orange or red dwarf. The smaller dwarf will occasionally expel a shell of hot gasses that it no longer needs or wants. These gases that it expels will be replenished from its larger star and will start the process over again. The star can increase in magnitude to over a million fold.
A Dwarf Nova is when the two starts that are close together are two white dwarfs instead of one being a dwarf and the other being a giant. They feed off one another thus replenishing their fuels.
Symbiotic stars are like the other novas except that instead of one being a white dwarf, the stars are a red giant and a blue hot star. These stars are held tightly together in a gas cloud typically.
These variable stars are ones that sudden expand and contract. Their visibility and brightness is due to their extremely large solar flares which alter their brightness.
These eruptive variables are slow to build up over time and their maximum brightness is sustained for a while before then falling back to their natural size.
These stars are low in Hydrogen and high in other elements which makes them near the end of their lives because star burn hydrogen to helium etc. etc all the way to Iron when they can't do anymore after that. A star of this type will expel its use materials which later will condense and thereby blocking out its light thus giving the impression of pulsating. The waste will eventually be expelled but not straight away.
These are binary stars where the darkness of one cancels out the light from the other as one rotates round the other. The most well known example of an eclipsing star is that of Algol which is found in the constellation of Perseus. Another group of eclipsing is the Beta Lyrae where the brightness of a star is eclipsed by a larger star but whose view has been obscured by an accretion disc of waste that the other star no longer needs. An example of this type of star is Sheliak in the constellation of Lyra. W Ursea Majoris is the third group of these stars where the eclipsing period is very short and consists of two similar stars, As their sizes are similar, they cause each other to be not spherical but tear-shaped.
Pulsar are neutron stars that rotate very very fast. They are created at the end of the death of a star once it has gone supernova. All what is left is a small neutron core than rotates/spins very fast. The Neutron star will fire gamma ray beams off into space and because the beams are not in all directions then it gives the impression it is variable and pulsating.
Wikipedia will have detailed information on all these types of Variables but if you looking for further brief information, then CSIRO and David Darling have more information.