Hipparchus sorted the stars he could see into six ranks. The brightest were of the first rank and the ones barely visible were of the sixth. It was a rough scheme with nothing measured, and it has stayed in use for two thousand years.
When the brightness was finally measured, the first rank turned out to be close to 100 times the sixth. So the scale was fixed on that: a difference of exactly 5 magnitudes is a ratio of exactly 100.
One magnitude is then 100^{1/5}, about 2.512 times. Written out, the rule is m_2 - m_1 = 2.5 \log \dfrac{F_1}{F_2}, where F is the amount of light arriving.
The one thing to hold on to is that a smaller number means a brighter object. Anything brighter than the old first rank runs into zero and past it into negative numbers.
Laid out on one scale, the sun is -26.7, the full moon is -12.7, Venus at its best is about -4.4, Sirius is -1.5 and Vega is 0.0. Vega sitting at zero is not a coincidence: the scale was tied to it.
The naked eye gives out at about 6. A large telescope reaches about 27. From the sun to there is nearly 54 magnitudes, a ratio of more than 10^{21}, and one scale carries all of it.
That is what a logarithmic scale is for. The eye itself responds to ratios rather than to differences, so a scale built on ratios matches what the eye does.
But every number here is the amount that arrives. It says nothing on its own about the star. A faint star nearby and a fierce star far away can land on the same mark.
To compare the stars themselves, the distance has to be taken out. The way it is done is to ask what the magnitude would be if the star were moved to a fixed distance.
That distance is 10 parsecs, and the magnitude there is called the absolute magnitude, written M. The apparent one is written m.
The sun has an absolute magnitude of about 4.8. Moved out to 10 parsecs it would be a star at the edge of naked-eye visibility, which is a fair measure of how ordinary it is.
Absolute magnitude is another way of writing luminosity. The vertical axis of the HR diagram can be labelled either way, and often is.
Since the only difference between m and M is where the star was put, the gap between them holds the distance and nothing else. It is m - M = 5 \log d - 5, with d in parsecs. The gap is called the distance modulus.
Check it at the fixed distance. At 10 parsecs the gap is 0, as it has to be. At 100 parsecs it is 5, at 10000 parsecs it is 15.
There are three quantities and one equation, so any two give the third. With the distance known from parallax, M follows, and that is how the main sequence was calibrated in the first place.
Run it the other way and it becomes a way of measuring distance. Find something whose M is known from its behaviour, measure its m, and the gap gives d. Every rung of the distance ladder above parallax works this way.
Spreading a star out into a full spectrum takes a long exposure. For temperature alone there is a shortcut: measure the magnitude twice, through two different filters, and take the difference.
The usual pair is B, which passes blue light around 440 nm, and V, which passes a band around 550 nm close to what the eye is most sensitive to. The difference B - V is called the color index.
A hot star has its curve still climbing at the blue end, so it delivers more through B than through V and B - V comes out small or negative. A cool star is the reverse and B - V comes out large. Vega is defined to sit at 0.
This is what the horizontal axis of a real HR diagram is. It is drawn as B - V rather than as temperature, which is why the axis runs from hot on the left to cool on the right rather than the other way round.