Why are capacitors use in transistor cicuits? What does it do with biasing operations? I always see capacitors on transistors being used as amplifiers.
A capacitor blocks DC, so it can be used to pass a signal (e.g. audio, etc) without it's DC level interfering with the DC bias of a transistor. This way the DC offset of the input signal can be at any level and the transistor amplifier will treat it the same way.
For example, if you have one transistor with it's collector output at a DC level of 5V, and the next transistor stage has it's base biased around 1V, directly connecting them would turn the second stage full on all the time as the input voltage will always be too high.
Take this simple circuit as an example:
Here are the waveforms at the various points:
Notice how the input voltage is a 1kHz 10mV signal with a 10VDC offset. After the input capacitor the DC level is now ~870mV. The same can be seen at the output cap. The gain is around 20 (200mV / 10mV = 20)
Now if we remove the caps:
And look at the input/output waveforms:
Things have changed drastically - the transistor base is now at 10VDC, which means the emitter will be at around 8-9V (the usual 0.7V drop will be higher due to excessive current) and the emitter resistor current will be ~8.5V / 100 = ~90mA. The output is stuck at around 20mV higher than the emitter. The circuit has no gain.
Apart from coupling, capacitors are also used for things like emitter bypass (as Vlad mentions) Here is the first circuit again with an emitter bypass capacitor added:
And the simulation:
Notice the gain has increase to around 100 from the first circuit (1V / 10mV = 100) This is because the emitter resistor provides negative feedback and controls the gain in the first circuit. With the capacitor added, the DC is unaffected but the AC now sees a lower impedance path to ground (the capacitor) so the AC gain is increased. So the AC is "bypassed" to ground.
There are many other uses for a capacitor, but these are the main uses in a typical audio amplifier circuit (apart from power rail filtering of course)