Bode (pronounced BOdee) plots are graphs that show how the magnitude and phase of an electric circuit's
system function vary for changing frequency. When done by hand, bode plots are best done as approximations to determine useful properties of the circuit. Mathematical programs like
MATLAB can do perfectly accurate bode plots, and often have standard functions built in to do so, but this is hardly ever necessary.
The bode plot of the magnitude is very useful for determining what kind of filter a circuit is. Even more importantly, it can help to determine exactly what the frequencies are when the circuit crosses certain important voltage thresholds. For instance, when the magnitude is equal to 1, the input voltage is equal to output voltage. An interesting example of the bode plot of a system function's magnitude is that of the notch filter:

H 
 
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omega
For frequencies around the middle, the magnitude drops off sharply. For all other
frequencies not near that middle value, the magnitude is high. Thus, the notch filter will filter out all values near that frequency. What frequency that is depends on the parameters (
resistances,
capacitances, etc.) of the circuit. You don't need a bode plot to tell you all of this information, as it can be done numerically. But it is nice to have and makes it easier to visualize.
How do humans make bode plots? There are key frequencies to mark down and solve for, and once that is done, it's easy. Simply solve for what happens at low frequencies, at high frequencies, and at the resonant frequency. You can draw sketches of the curve at those points, and connect the estimates to get a pretty good graph.
Many phase plots tend to look similar, having a shape like the function y=  x^{3}:

phase
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The time delay changes a lot around the middle of the plot, and flattens out at values not close to that resonant frequency.
Because of the varying values of frequency and the response they get from certain circuits, it is often prudent to use a logarithmic scale on at least one of the axes instead of a straight numerical scale. If not, you will end up with a long, stretchedout graph that won't offer much information.