A buffer is a circuit that provides unity gain, a very high input impedance, and a very low output impedance. The rule of thumb is that most (but not all) guitar circuits should have a high input impedance and a low output impedance in order to maintain a pristine signal (no tone suck). Unfortunately, the reason this matters brings us in direct contact with basic electronic theory, so in order to understand buffers and use them intelligently, we have to think about electronics instead of just thinking about 'tone'. There will even be a (very) little bit of math. Sorry about that. But if you're really interested, here you go. (If math really bugs you, just read past it. You'll probably still understand the conclusions.)
This rule is based around the concept of
voltage dividers. The output voltage of a voltage divider is Vout = Vin * Z2/(Z1+Z2) (look at the link for what Z1 and Z2 refer to). The junction between two circuits is modeled as a voltage divider, where Z1 is the output impedance of the first circuit (which can be a pedal or your pickups) and Z2 is the input impedance if the second circuit (which may a be a pedal or your amplifier). If Z1 is much, much larger than Z2, that equation implies that Vout is roughly equal to Vin. Plug Z2=1,000,000 ohms and Z1=1 ohm into the above equation and see what you get.
The definition of impedance is Z = V/I, where V is voltage and I is current. In less mathematical terms, impedance limits current. Low output impedance means that the circuit will provide lots of current. High input impedance means that the circuit doesn't require much current. So the combination of low output impedance feeding high input impedance implies that the first circuit will not be 'loaded down', ie. that circuit will be able to produce enough output current that the output voltage (the signal) will not be dropped by the current consumption of the next circuit. In practice, that means that the first circuit will produce the same output voltage when it is not connected to anything (infinitely small load), as when it is plugged into another circuit. If a circuit has a high output impedance, and feeds a circuit with a low input impedance, then a circuit that can't produce much current is trying to supply a circuit that requires a lot of current. So the first circuit can't keep up with the demand and it's output voltage drops. Plug Z2 = 1000 ohms and Z1 = 10,000 ohms into the voltage divider equation and see what you get.
There is a range of acceptable input and output impedances, but 'high' Z is usually somewhere around 1M-10M ohms, and 'low' Z is generally less than 1k ohms or so (often much less for a good buffer, maybe 1 ohm -100 ohms, not including the output cable).
That's why they are called 'buffers'. A buffer is a separation between two things. By buffering the input and output of a circuit, you are essentially isolating that circuit, ie. making it so that the output voltage of the circuit depends only on the input voltage, not on the output impedance of the previous circuit nor the input impedance of the following circuit. Mathematically, by making the input Z high and the output Z low, you are making Vout almost equal to Vin, no matter what the impedances of the surrounding circuits are (within reason).
Buffers are often included at the input and outputs of commercial pedals, and sometimes remain connected in bypass (Boss, etc.). Buffers make it so that you can run very long cables without affecting your tone, for exactly the reasons described above. Conceptually, consider that guitar cables can be modeled as capacitors to ground with capacitances of 10pF - 50pF per foot, and that capacitors have high impedance at low frequencies, and low impedance at high frequencies. So by using a buffer we can keep our tone pristine (Vout ~= Vin in the voltage divider equation) because, even though Z2 gets smaller for high frequencies when the cable gets very long, Z1 is very, very low. You can also use the
low-pass filter equation (which is derived from the voltage divider equation) with R set very low to find that the corner frequency (the frequency above which the cable drops amplitude from the signal) is higher than the frequency range of the guitar, even when the cable is very, very long.
The downside of buffers is fairly small, and generally only applies to two situations. First, some circuits (particularly those with bjt first stages with low or non-existent emitter resistors, like the Fuzz Face, LPB-1, etc.) are designed to be fed by guitar pickups, so they expect to be fed a signal from a source with a fairly high output Z. One way of thinking about this is that these circuits are not supposed to be isolated from the pickups, ie. the pickups are
part of the circuit. When the source impedance is too low, the gains of these circuits are too high, and they behave poorly (they sound shrill because the high frequencies are expected to be attenuated and the buffer prevents that, and the expected volume knob cleanup disappears).
The other downside comes from too many stacked buffers. If Vout = Vin * 0.99 with one buffer, the volume drop is unnoticeable. But after 10 buffers, Vout = Vin * 0.9, and the volume drop will start to become noticeable. If the buffers are not optimal (bjt buffers are less effective than op-amps, for example) the volume drop will get worse. Each buffer also contributes a little bit of noise, so if you try to use a boost to get the volume back, you will also amplify that noise.
So your use of buffers will depend on your rig and your goals (some guitar players, like Jimi Hendrix and Brian Setzer, used really long cables to drop some of high frequencies before their signal hits their amp). But the general rule should be something like: not too many buffers, and not before Fuzz Faces, COT-50's, LPB-1's, etc. Otherwise, put one as close to your pickups as possible, unless you are trying to bleed off high frequencies without using the tone knobs on your amp for some reason. Figure out which of your pedals have buffered bypass, and recognize that buffers only impact the part of the signal chain that directly follows them, so buffering the input of a pedal generally has no impact on anything that comes after the pedal when the pedal is engaged (or if it has buffered bypass). But most pedals are designed to have low output Z, so you shouldn't need an extra buffer after any pedal, unless it's poorly designed. Likewise, a buffer can easily drive a 300 ft cable, but a 300 ft cable followed by a buffer will sound dark because much of the treble has been bled to ground before the signal reaches the buffer.
If you're still reading, I hope this was helpful.
Locked
I'd kinda had the vague impression that whas what they did, but it always helps to have these things confirmed and the clear and informative answers have certainly given me a much better idea of the details 
