What is the purpose of a bias resistor?

If your device is powered from a single voltage and ground, it can't output anything below ground. In order to reproduce a signal that varies between +V and -V, you need to shift it upwards so that it varies from 0 to +2V instead. The DC offset is the bias.

http://en.wikipedia.org/wiki/Biasing

The bias is the operating point.

For a bipolar transistor (BJT) the bias resistor will maintain enough current into the base so that the transistor is neither saturated (fully on) or cut-off (fully off).

Some BJTs come with an internal bias resistor to reduce the parts count in a design. If you are switching BJTs on or off you don't need a "bias" resistor but you may need a resistor to limit the current into the base.

In analogue working, transistors (and before that, valves (or vacuum tube devices)) do not have a fully linear response, ie the output is not exactly proportional to the input over the full operating range. If you are wanting a linear response, you move the input signal into the middle of the linear part of the operating range by using bias resistors (and you restrict the input signal such that it does not go outside the linear range).