12AX7 is a nine pin miniature, twin triode tube.
Twin triode means it has two separate tubes inside one glass envelope.

Each triode has three electrodes: plate, grid and cathode.
At pin 1 is the Plate or Anode.
At pin 2 is the Grid.
At pin 3 is the Cathode.
The other triode is pins, 6-Plate, 7-Grid, and 8-Cathode.

There is a heater filament between pins 9 and 4, and another between pins 9 and 5. The heater circuit is often omitted from circuit diagrams, it is not considered to be a ‘working’ electrode as it plays no part in the audio circuit. To wire the heaters for 6.3 volts (heaters in parallel) you tie pins 4 & 5 together.

The Plate is connected to the high voltage B+ through the Plate load resistor.
The Cathode is connected through a Cathode resistor to ground.
The Grid is where the AC signal comes in, a Grid stopper resistor is usually connected here.

The first electrode is called the cathode and it gives off electrons easily when heated.  Near the cathode is the heater filament, pass current through the heater so that it gets hot and the cathode also get hot.

A second electrode called the anode is added, if the anode voltage is made positive relative to the cathode, it will attract electrons from the cathode and current will flow. If the anode voltage is made negative relative to the cathode, electrons will not be attracted to it and no current will flow. Being a one-way device with two electrodes, the tube is called a diode.

The current flowing between the anode and cathode is called the plate or anode current (Ia).  The voltage measured between anode and cathode is called the plate or anode voltage (Va).  The supply voltage is called the B+ or HT. 

When anode voltage is low it has a low influence over anode current, as anode voltage is increased the anode becomes more effective at drawing current from the cathode, until drawing all the current the cathode is capable of is called saturation.  As the B+ is increased the current in the diode will also increase, so for every value of anode voltage there is a corresponding value of anode current.  These points are plotted on the static anode characteristic graph.  

A third electrode, called the control grid, is added between the anode and cathode. With three electrodes the tube is called a triode.

If the grid is made positive relative to the cathode, electrons from the cathode will be drawn towards its electric field, but because the grid is very fine most of them will fly straight through the gaps and be captured by the electric field of the anode.

If the grid is made negative relative to the cathode, it will repel electrons, they stay at the cathode, restricting anode current. If it is made very negative then the anode current will be cut-off altogether.

Because the grid is much closer to the cathode than the anode, a small change in grid voltage has a much more powerful effect on anode current than a similar change in anode voltage

For every value of anode voltage there is now a whole range of anode currents. These are plotted as grid curves on the static anode characteristic graph.
If we apply a varying audio signal voltage to the grid then current in the valve will vary too. By putting a resistance (plate load resistor) in series with the tube as the current changes, the voltage drop across this resistor changes, generating a corresponding audio voltage at the anode.

To shows us any possible voltages and currents in the circuit, we will draw a load line on the static anode characteristics chart. Anticipating the B+ voltage supply to the preamp tubes, at node C, to be around 270 volts and using a plate load resistor of 91K.

If the tube is cut-off so that no anode current flows across the plate load resistor there will be no voltage drop, so the full B+ (270volts) will reach the anode.
This point can be plotted on the graph:
Va = 270v, Ia = 0mA.

With the full B+ dropped across the plate load resistor and none across the valve, using Ohm's Law, Voltage / Resistance = Current, the anode current would be 270v/91k = 2.96mA.
This point can also be plotted at:
Va = 0v, Ia = 2.96mA

Draw a straight (purple) line between the two points, completing the load line.

For any grid voltage value along the load line, there is a corresponding value of anode voltage and anode current. These are the only combinations that are possible in this circuit.

A small change in grid voltage causes a change in anode current, which causes a large change in anode voltage. Where the -1v grid line intersects with the load line there is 154 volts at the anode. Where the -2v grid line intersects with the load line there is 211 volts at the anode.

The grid voltage has changed by 1 volt but the anode voltage has changed by 57 volts. The gain of this circuit is 57/1= 57

Cathode biasing is when a resistor is placed in series with the cathode and a voltage develops across it, raising the cathode voltage.
Instead of making the grid negative with respect to the cathode, the cathode is made positive with respect to the grid.

To choose a resistor that will set the cathode voltage to 1.5 volts, it will set the grid voltage to –1.5v.
On the –1.5v grid line the Current is .95mA. (1.5 volts / .95mA = 1.57K). The Cathode resistor needed is 1.57k.
For a -2v grid voltage bias point the graph indicates that Current is .65mA. (2 volts / .65mA = 3K). The Cathode resistor needed is 3k.

Suppose that I have a 2.4k resistor I want to try. Voltage/Resistance = Current. 2 volts / 2.4k = .83mA . Mark this point on the -2 volt grid line. 1.5 volts / 2.4k = .62mA. Mark this point on the -1.5 volt grid line.  Draw a (green) line between these two points where this line intersects the load line will be the bias point.

Adding a cathode resistor to the circuit will alter the load line, the cathode and plate load resistors are now both in series with the valve.
Because the value of the cathode resistor is so small it will not make much difference. 270v/(91k+2k) = 2.90mA.

At -1.8 volts the bias point is set close to centre biased with the anode voltage being around 201 volts. The AC input signal voltage swings symmetrically around this bias point, from the bias point the up going and down going voltages are equal. It swings up 1.8 volts to 0 volts and down 1.8 volts to -3.6 volts.
But the anode voltage swings asymmetrically around this bias point. The down going voltage is (111volts) 201volts down to 90volts on the 0 volt grid line, and the up going voltage is (69volts) 201volts up to 270 on the -3.6 volt grid line, the full B+.
The output waveform is bent out of shape (distorted) and no longer a perfect sine wave. This bending of the waveform introduces new frequencies into the audio signal. These new frequencies are called harmonics and are integer multiples of the original audio frequency. Triodes, therefore, can have a pleasant warming effect on tone because they produce a decaying series of all harmonics, dominated by the second. Because the waveform is compressed on one side and stretched on the other, it is no longer equal above and below the bias point. The average value of the output waveform is not at the bias point, but has an average DC offset that can shift the average operating point of the tube dynamically with signal level, having a compressor like effect.

Calculating Second Order Harmonic Distortion = (AB-BC/2x(AB+BC)) x100.
AB is the difference between A & B(111volts), BC is the difference between B & C.
A=Plate Voltage where grid Voltage = 0,   B=Plate Voltage at Bias Point,    C= Maximum Plate Voltage. 

An AC sine wave swings both positive and negative relative to an average grid voltage or bias point. If a large enough input signal is placed on the grid, the triode will be overdriven into either anode current cut-off Clipping or grid current Clipping.
Drive the grid negative enough and the tube will reach cut-off and stop conducting. ( no Current will flow) The grid voltage may continue to swing more negative, the tube will remain in cut-off and the top of the output wave will be clipped. Biasing a tube closer to cut-off results in less power dissipation and is called cool biasing.

When the grid voltage approaches the cathode voltage, electrons being drawn from the cathode get attracted to the grid rather than the anode causing a forward grid current into the grid. This current causes a voltage drop across whatever resistance happens to be in series with the grid, making it hard to drive the grid past 0V. The more we attempt to make the grid positive the more current flows into it and prevents us from doing so. It is actually the grid signal that is being clipped, while the tube continues to amplify what appears on its grid normally.  Grid current limiting(bottom of the output waveform is clipped) occurs when we try to drive the grid positive, beyond 0v. The more resistance in series with the grid the greater the voltage drop caused by this grid current, and the harder and more abrupt the clipping.  Biasing a tube closer to Grid current Clipping results in more power dissipation and is called warm biasing.  Centre biasing offers maximum headroom before the signal is clipped.

A Static Anode Characteristic Graph with a 91k and a 120k load line.

The difference between plate load resistors is the smaller load resistance produces a steeper line and more harmonic distortion. While a larger load resistance will increase gain and voltage swing.

91k load line, B+ 270v.
Total Tube Series Resistance 95k, 270v/95K = 2.85mA
Gain 210v-152v = 58
Cathode resistor = 1.75v / .8mA = 2.2k.
Voltage Swing = 270v-88v = 182v
Bias @ 196 volts
Second Order Harmonic Distortion = 108-74/2x(108+74x100 = 9.3%

120k load line, B+ 270v.
Total Tube Series Resistance 125k, 270v/125K = 2.15mA
Gain 203v-140v = 63
Cathode resistor = 1.75v / .67mA = 2.6k
Voltage Swing = 270v-72v = 198v
Bias @ 187 volts
Second Order Harmonic Distortion = 115-83/2x(115+83)x100 = 8.1%

page info resources from a highly recommended book Designing Valve Preamps for Guitar and Bass By The Valve Wizard