Magnitude systems
Look at alpha and beta Centauri, the "pointers". Beta is definitely bluer than alpha. And it appears 0.64 magnitudes fainter. We will see later that the colors differ because the surface temperatures of stars range from 4000K on up and the hotter the star the bluer the star. But the eye peak response is at 555nm when light adapted and 515nm (green) when looking at stars. (This change in eye response with dark adaptation is related to the Purkinje effect.) So what exactly is a magnitude? The answer is really complicated but one that you should get right. First, if the apparent brightness of a star is measured by eye or some optical recording system that mimics the human eye response (filter plus photocell or photographic medium for example) the magnitude is called visual apparent magnitude m_vis or m_v. The notation m_vis is preferred. Photographic plates (eg Eastman 103a-O) are rather more sensitive to blue light and not sensitive at all in the red so bCen "looks" as bright as aCen. We call such derived magnitudes photographic magnitudes m_pg. If the film is "panchromatic" to approximate the human eye response, it is still different enough that we label the magnitude photovisual, m_pv. And there's more...
Wouldn't it be better to just correct all measurements to a value that one would get with an ideal detector, one that detects all wavelengths with the same efficiency? Such a detector is called a bolometer and such magnitudes are bolometric magnitudes m_bol, highly desirable but very difficult to obtain in practice. One usually converts observed magnitudes to bolometric via a model but in fact it is the model we are trying to derive by observations in the first place.
Finally, in order to accommodate the effects of detector spectral sensitivity, a standard system was proposed by Johnson and Morgan (1953) who defined specific detector sensitivity curves by means of filters. (Ideally the filters are constructed to match the equipment used so the response is the same for all observers.) The original system used filters in the (U)ltraviolet, (B)lue and (V)isual regions of the spectrum and has been extended to the (R)ed, (I)nfrared and beyond (J,K,L...) by Stebbins and Kron (1956). Such apparent magnitudes would logically be labeled m_U, m_B etc. but by convention are labeled U, B, V, R, I... and the system is called the Johnson UBVRI and Kron RI systems (covers λ310-λ900nm).
Stromgren has developed a slightly different (more appropriate) four-color system, uvby (ultraviolet, violet, blue, yellow). The y is very nearly the same as the V magnitude.
But wait, there's more! Cousens added some colors to the UBV and so now we have UBVRrIiJHKL.... (small letters are Cousins) and other astronomers have devised still more systems. One big job in astronomy is trying to find ways to intercompare the different systems!

In this class we will concentrate on UBV. Most star catalogues give V, often B-V and sometimes U-B.

Color indices
The UBV data are quite informative. The difference between any two colors is called a color index and if you hear a astronomer say "color" it usually means B-V. Stars like the sun and aCenA (alpha centauri is actually a double star, "A" slightly more massive than the sun and "B" somewhat less massive) put out rather more light in the V than the B so the V magnitude is less than the B magnitude (see, magnitudes go backwards) and B-V is positive. Sirius, almost twice as hot as the sun, has B=V and really hot blue stars have negative B-V. The B-V color index ranges from -0.4 for infinite temperature to over +2.0 for cool red stars. In fact, B-V tells you the surface temperature of a star right away.

Absolute magnitudes and distance modulus
The color of a star tells us how hot it is but how bright is it? The brightness depends not only on temperature (to the fourth power) but also on radiating surface area or radius squared. We define absolute magnitude to be the brightness in a given system that an object would appear to have at a distance of 10pc. I would have used 1 pc but 10pc is what is used. The difference between apparent and absolute magnitude is the distance modulus DM and is just

From the inverse square law l(10)/l(r)=(r/10)² so DM=m-M=5log(r/10).

And the distance in pc is just r=10x10^0.2DM=10^1+0.2DM.

It is usual to denote absolute magnitude by M so the absolute V magnitude would be M_V and the most prized magnitude of all would be M_bol.
Note that the (*intrinsic) color index does not depend on distance. The CI is not a magnitude, it is a magnitude difference or brightness ratio and the ratio is independent of distance.
Galactic astronomers often work in distance modulae rather than megaparsecs. (*Interstellar reddening will increase the observed CI as we will see later on.)
Finally note that there is a difference in the "units" associated with m and M. Apparent magnitude is a measure of the flux [Watt per square metre, Wm^-2] hitting the earth while Absolute magnitude is a measure of the star's intrinsic luminosity [W]. So I can tell you what the absolute bolometric magnitude of a 100W light bulb is but the apparent magnitude depends on how far away it is, the DM = m_bol-M_bol.

Magnitude zero
Now for the tricky part. You may be familiar with sound intensity which is measured in bel or decibel (dB). The bel is a logarithmic quantity like the magnitude but it is base ten and increases with intensity (just like the magnitude system should have been). the zero point is set at 10^-12 Wm^-2. Very neat. A sound level of 85dB is 10^-12+8.5 Wm^-2.
Astronomers have set up a set of standard magnitudes based on a set of stars known as the north polar sequence, and attempted to make U=B=V for stars of spectral class A0 V, and Vega (αLyr) almost does this: U=+0.02, B=V=+0.03. Note that this is done for convenience of the observer who measures magnitudes by observing the ratio of the object brightness to a standard star. Astronomers do not have instruments calibrated in magnitudes like acoustic engineers have "dB metrs". And each magnitude tyoe has its own arbitrary zero point!

Here is one way to solve the problem of the zero point. (Warning, this is not official.) IAU Commission 25 (1997) XIII General Assembly recommended the bolometric magnitude of the Sun be defined +4.75 (other values in common use are +4.74, Allen, A.Q. 1999 and I have seen +4.72 mentioned).
The best value for the mean solar flux (we call this the luminosity is 3.84216X10²⁶ Watt (from the measured mean insolation of 1366.2 W/m², which varies between 1365 and 1369 W/m² over the sunspot cycle) which implies zero bolometric magnitude corresponds to 3.052X10²⁸ Watt. A hard nosed physicist probably would have said "why not make magnitude zero be 3X10²⁸ Watt" -- what would that make the bolometric magnitude of the Sun? (+4.731)