In short, Dwarf Stars are normal, main sequence stars compared to the giants of the Universe such as UY Scuti. The term Dwarf Star was created by Ejnar Hertzsprung to distinguish between the two different types of Red Stars, those that are giant stars such as Antares and those that are much smaller such as Proxima Centauri. Hertzsprung who along with Henry Norris Russell created the famous Hertzsprung-Russell star size map.
The star map shows the Sun as being in the middle, the sun is classed as a Yellow Dwarf star because although it is one of the larger stars in the galaxy, it is tiny to compared to UY Scuti, the currently recognised largest star in the milky way which when you discover that UY Scuti is over 1,700 times as big as the Sun, you can see why the Sun is called a Yellow Dwarf star.
In the below diagram, you'll have noticed the green strip. That strip should be white, there are no green stars. Stars don't shine exclusively in one colour and as green stars are in the middle, the colours to the side cancel out the green and therefore the light appears white. If there were green not white stars out there, our star, the Sun would be a green star.
Dwarf Stars would at the bottom of the diagram with the giant stars at or near the top.
When it comes to size of a star, size matters, it is best to be small rather than large because the smaller stars convert their hydrogen to helium at a slower rate than larger stars and will live longer. Our Sun is expected to die in about 5 billion years, that's so long in the future, we don't need to worry about. When we move and if we ever do move to another planet, we should find a planet orbiting a red dwarf as they will last for possibly trillions of years.
When stars use up their hydrogen, you'd think that they'd begin to shrink, instead the opposite is true. As a star uses up it fuel, the outward pressure overpowers gravity and they expand outwards. Betelgeuse was probably a normal star until it finished its main sequence and then expanded outwards to the size it is today.
A Dwarf Star will generally have a spectral type that V, VI, VII. There aren't that many stars that have VI or VII spectral type, most dwarf stars will just be V without any I beforehand.
Dwarf planets are just as suitable as larger stars to support life. For a Sun-sized star, the planet would need to be about the same distance as Earth is to the Sun for it to support life. If the star is smaller, the planet would need to be nearer the star for the planet to be hot enough for life. If the planet is large, the planet would need to be further out from the star to support life.
The nearest Dwarf Star to us is Proxima Centauri which is just over 4 light years away. It would take light over four years to get to us if it left today. The star is not visible to the naked eye, you would need some optical aid such as telescope to see it. The star has an orbiting planet but there's no chance of life as the star is a Flare Star meaning it explodes with plasma which has the power to strip away the planets atmosphere.
White Dwarf stars are the end of life result of an old star. If the star is not massive, it will end its days as White Dwarf. A White Dwarf will eventually burn its self out and no longer light up. A White Dwarf will end itself as a Black Dwarf star.
Due to how long it takes to transition from a White to Black Dwarf star, it is believed that the Universe is not old enough for a White Dwarf to exist yet.
Brown Dwarf stars are a form of failed star. They are too big to be a gas giant but not big enough to start thermonuclear reaction to turn hydrogen into helium. These are hard to spot because they are so dark against the blackness of the universe. There are only a few so far discovered such as
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