An empirical relation used in astronomy to link a galaxy's mass with it luminosity. The Tully-Fisher relation is most useful in that it then enables a person to roughly estimate the distance to the galaxy. The Tully-Fisher relation is least useful in that its empirical nature requires astronomers to calibrate it using known sources. Known sources are (generally speaking) both rare and unreliable in the world of astronomy, and the exact values of the calibration constants are still under debate.
The train of logic behind the relation says that the faster a galaxy is spinning, the higher its luminosity. We can relate how fast the galaxy is spinning to its mass by a simple centripetal force calculation. We can find how fast the galaxy is spinning by looking at its spectral emission (to be explained below). Put all this together, and you get mass --> spin rate --> luminosity. You can then use this to find distance using some formulas that are a liiiiiiittle bit beyond me right now as a relative newbie in the astronomy world.
Because the galaxy is spinning, an observer sees part of the galaxy as coming towards him and part of it as moving away. The Doppler Effect causes light being emmitted from the part of the galaxy that is spinning towards the observer to appear blueshifted (demonstrating an artificially decreased wavelength), and light being emitted from the part of the galaxy that is receding to appear redshifted. This widens the spectral line of the galaxy. The wider the line, the more the galaxy is spinning.