Bias affects:
* the amount of amplification. More amplification means taller sound waves. I.e., smooth sine waves approach the shape or rectangular waves: steep walls approaching 90°; shoulders approaching square instead of gently rounded; tops approaching flat. Changes in the wave form = harmonic distortion which changes the tonal characteristics of the signal. At some point, simply adding amplification makes the tone more "clear" then more harsh. Depending upon bias setting & incoming signal strength, the wave will eventually "hard clip" -- the wave's peak literally shaves-off and becomes flat -- going from "clean" to a soft over drive tone, to a harsher form of distortion.
Further to what jjasilli said
In an inverting gain stage, you can bias hotter or cooler, and this changes the type of clipping in different ways, which can affect the tone.
Biasing hotter eventually causes grid-current-limiting, where the positive voltage swing of the grid is clipped. This type of clipping is also affected by the source impedance of whatever circuit precedes the stage in question. With grid current limiting, it is the grid signal that is being clipped, and the plate continues to amplify the grid's already-clipped signal. This type of clipping can have rounder edges* where the signal enters and exits clipping, which gives a bluesy flavour to the tone.
* affected by the source impedance of whatever is prior to the stage in the circuit.
Biasing cooler causes cutoff at the plate which is a different type of clipping. In cutoff the grid isn't being clipped, but the bias is so cold that when the grid voltage swings down to its bottom 'peak' there is no more voltage for the plate to go 'up to' because the plate is effectively at HT voltage. So the signal at the plate becomes clipped. This type of clipping typically has 'squarer' edges where the signal enters and exits clipping, and produces a more 'bitey' rock tone.
In many hi-gain amps they set up the bias of successive pre-amp gain stages in order to alternately have both types of clipping. 1st hot, 2nd cold, 3rd hot, 4th cold etc to get a 'sustainy' tone.
This is all explained in great detail in Merlin Belcowe's 1st book BTW.
http://shop.dobermanamps.com/product-p/book-dvpgb.htm