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The Hertzsprung-Russell diagram is a tool that astronomers and astrophysicists use to classify different types of stars. It is sometimes called the Color-Magnitude Diagram (CMD).
The Hertzsprung-Russell diagram primarily plots color/heat of a star against its absolute magnitude/luminosity. Color and heat are the x-axis. They share an axis because color and heat correlate very well with each other. Up to about 3500 K surface temperature, stars are red, up to about 4500 K they are orange, up to about 6000 K they are yellow, up to around 9000 K they are white, and beyond that are blue. Some stars, such as Wolf-Rayet stars, have surface temperatures up to 25,000 K.
Absolute magnitude/luminosity are the y axis of the Hertzsprung-Russell diagram. Again, like with the x axis, the y axis can share two variables because the two are correlated. Absolute magnitude is measured using a standard luminosity scale, while the luminosity side is measured in terms of solar units. The solar units side is a logarithmic scale, because stars range in their luminosity from as much as 1/100,000th the Sun to ~100,000 times the Sun. During a supernova, a star shines as brightly as five billion times the Sun, although this is only temporary.
The Hertzsprung-Russell diagram shows that stars cluster into several natural categories: the main sequence (dwarfs), of which our Sun is a member, consists of hydrogen-fusing stars; white dwarfs, which have exhausted their nuclear fuel and are in the process of cooling down; and the subgiants, giants, and supergiants, which fuse together elements heavier than hydrogen, such as helium.
White dwarf stars are much smaller than the Sun: they have a diameter only similar to that of Earth! This is because a white dwarf is a remnant of the core of a star that previously generated its own energy. The dwarf stars are much larger, around the size of our Sun. The giants are much larger still, as large as the orbit of Mars. This is because they fuse their nuclear fuel more rapidly, heating them up and expanding their envelope.
The standard name for stars comes from the Hertzsprung-Russell diagram. For instance, if a star is red and in the supergiant class, it is called a red supergiant. Our Sun is a white dwarf. There are yellow giants, blue giants, orange dwarfs, red dwarfs, and many other known star categories.