What is the basis for the stellar magnitude scale? Is one star, like Polaris or Sirius taken to be 1.0 and the rest compared to it?

The stellar, apparent magnitude scale was first used by Hipparchus and then later by Ptolemy in the second century A.D. It was qualitatively defined such that the 20 brightest stars were identified as the stars of the 'First Magnitude', and the faintest stars you could see, as stars of the Sixth Magnitude. Stars in between were subjectively assigned to the intermediate magnitude ranks, 2, 3, 4 and 5.

Nothing more was done with this system until the middle of the 19th century when Pogson at Oxford, in 1854, quantified the scale by observing stars with a large telescope (Herschell's) and a small telescope and noting that the brightness of the Sixth Magnitude stars was about 100 times fainter than the first magnitude stars.

The upshot of this is that each increment of magnitude represents to the human eye a change by a factor of 100 to the one fifth power or 2.512 so that a Second magnitude star is 2.512 fainter than a First. A seventh magnitude star is 2.512 x 2.512 x 2.512 x 2.512 x 2.512 x 2.512 = 251 times FAINTER than a first magnitude star, and a 24th magnitude star, like the ones seen by the Hubble Space Telescope are 2.512^(24 - 1) = 2.512^23 = 1.58 billion times fainter than a first magnitude star. On this scale, the Sun has a magnitude of -26.0 which is 2.512^25 = 10 billion times BRIGHTER than a first magnitude star.

Normon Pogson assigned a magnitude of 2.0 to the two stars Aldebaran and Altair, and all other stars were referenced to these two in his system. By 1911, Polaris had also been assigned a magnitude of 2.0, but Einar Hertzsprung discovered that it was a variable star.

By 1906, using a prismatic device on a meridian transit telescope, E. Pickering at Harvard compiled a list of some 50,000 stars based on over 1 million comparative measurements. This 'Harvard Catalog' also helped to fix the zero point for the apparent magnitude scale by reconciling the magnitude estimates from several different observers into one single consistent catalog of brightnesses.

With the advent of accurate photographic films and electronic instruments to precisely measure the intensities of objects seen through the telescope, the apparent magnitude scale can be placed on an absolute scale of intensities. These techniques now require that you specify at which wavelength you are observing the object, since you can now place various specially-designed filters over the 'photometer' or 'photographic plate' which will only pass light in a very specific wavelength range. The naked eye roughly corresponds to 'V-band' between 4800 and 6800 Angstroms.

The Standard Star used for these photometric purposes is an A0 main sequence star which has a magnitude of zero and a B-V 'color index' of zero. The naked-eye star Vega which is an A0V star with a temperature of 10,000 K, has a photoelectric, V-band magnitude of +0.04. All other stars are referenced to this star from the ultraviolet through the near infrared covering the photometric filters U, V, B, R, I, J, K, L. The spectrum of Vega has been measured in terms of its absolute intensity compared to a laboratory 'black body' temperature source with a number of atmospheric corrections applied by astronomers J. Beverly Oke, Rudolph Schild, and Arthur Code back in the 1960s. So, the apparent magnitude of a star relative to Vega can now be translated into an absolute measure of its intensity in ergs per centimeter square per second per wavelength or frequency interval. At a wavelength of 550 Angstroms, a zero magnitude A0V star has a flux just outside the earth's atmosphere of 3.8 billionths of an erg per centimeter square per second per Angstrom. The relationship between flux and wavelength for a zero magnitude star can now be written in terms of the absolute intensity of the star in watts per meter square per Hertz. For example, if we were to use a K-band filter to isolate the light from a star near 2.2 microns in wavelength, a zero magnitude star would yield 630 Janskys, where 1 Jansky = 10^(-26) watts/meter square/Hertz. or 100 trillionth of a trillionth of a watt per meter square per Hertz.