What exactly are they measuring? How do they measure it? What are the implications of the different approaches on accuracy / speed of getting a good measure?
It would be helpful if someone could give an answer that explains how hub-based meters like the PowerTap differ from the crank-based meters like the Quarq which differ from the other ones.
There are several different types of power meter on the market and each measures something slightly different in order to make their estimates. In addition, the way that they measure what they measure has implications for their accuracy. Below I discuss what the major models measure, how they measure it, and the implications for accuracy.
Power is the rate of work (so you need to know the amount of work and the span of time over which that work is done), and work is a force exerted over a distance, so each power meter has a different way to measure those forces and, because of patents, each has chosen to measure them in a different "location."
With the exception of the iBike, most power meters measure the forces somewhere along the drivetrain: working from the back to the front, the PowerTap (and the old Look MaxOne) measures at the rear hub, the older Polar systems measured along the chain, the Quarq, SRM, Rotor, and Power2Max measure at the spider of the front chain ring, the new Look/Polar and Garmin Metrigear (thus far, announced but unreleased) measure at the pedal spindle, the (announced but unreleased) Brim Brothers measures at the shoe cleat, the Ergomo measured at the bottom bracket, and the Stages measures at the left crank. The iBike measures in a completely different way, discussed below. One consequence of measuring at various points along the drivetrain is that drivetrain losses will (or won't) be accounted for to a different degree; for example, a PowerTap will usually read lower than a SRM since one is "upstream" of most drivetrain losses while the other is "downstream." This difference is more of a definitional issue than a strict "accuracy" issue (in the sense of, "is gross income or net income a more 'accurate' measure of income?" Unless you have a specific use in mind, it's hard to say which is more "accurate").
Most of the power meters on the market use strain gages, which are small thin foil strips whose electrical conductance and resistance varies as they are deformed. Strain gages are used in lots of applications (like, bridges) and their properties are well-understood. In general, strain gages are combined in a "rosette" or "Wheatstone bridge" in order to produce more accuracy and precision (more strain gages usually produce better results), and, when operating properly the Power Tap, Quarq, and SRM are usually accurate to within a couple percent (and, just as importantly, with high precision); this has been verified both statically (using known weights hung from the crank) and also dynamically (using a large powered rolling drum in a lab). The forces are then combined with a measurement of angular velocity or speed to get power. A virtue of strain gages is that the change in resistance can be measured even when the device is stationary so the cyclist can measure the accuracy of the strain-gage-based power meters at home by hanging weights of a known mass from the crank. A common problem with the strain gage approach, however, is that they can be sensitive to changes in temperature and so need to be "re-zeroed" prior to (and sometimes, during) rides. The old Look MaxOne's achilles heel was waterproofness, not the strain gages or method of measurement. For example, the original Power2Max (and the old discontinued SRM "Amateur" model) uses fewer strain gages than the current PowerTap, Quarq, or SRM models and reports from users (subsequently admitted to by the manufacturer) showed that it was more sensitive to temperature drift during a ride than those others. The Power2Max was redesigned and updated in late 2012 and reports are that the temperature problem has been largely addressed. A claimed feature of the Stages is that it is designed around automatic temperature compensation — as of early 2013 this claim is still being evaluated by users and it is too early to know if their approach does what it claims.
The old Polar power meter measured the force transmitted along the chain by chain tension, and included a chain speed sensor to get total work. In a chain higher force transmitted along the chain results in higher tension, and tension can be measured by the resonant frequency of the object (for example, plucking a highly-tensioned spoke with your fingernail produces a high frequency tone while plucking a loose spoke produces a low tone). As an historical aside, the proof-of-concept prototype for the Polar chain tension sensor was the pickup from an electric guitar. The chain speed sensor fit on one of the derailleur jockey wheels and could count the "pulses" in the magnetic field as the chain rivets passed; since chain rivets are a known distance apart, chain speed was easily calculated. As for accuracy, when the Polar was operating well, it was very good; however, when it wasn't it was very naughty indeed. Worse, it was often hard to tell when it was being naughty. The downfall of the old Polar power meter was three-fold: 1) the chain tension sensor needed to be close to the chain, which was hard to achieve since the chain sometimes had to be in the large or small chain ring or the large or small rear cog; 2) the chainspeed sensor sometimes got overwhelmed and gave false speed readings; and 3) incomplete weatherproofness in part due to exposed wires and a poorly sealed "pod."
The Ergomo bottom-bracket based power meter used an optical sensor and a series of "peek holes" to measure torsion in the bottom bracket. An odd characteristic of this design is that it could only measure the (torsional) force traveling through the bottom bracket; thus, it only measured the power contributed by the left leg: to get total power it doubled the left-leg contribution. In conjunction with the difficulty in installing and calibrating the Ergomo (it had to be installed just exactly so), this dependence on bilateral symmetry between the legs was the death knell for the Ergomo. The Stages power meter similarly measures force by deformation in the left crank, and doubles the "left" to arrive at an estimate of total power. Research with instrumented force pedals shows that bilateral asymmetry in power production between the right and left legs is the norm — worse, the research shows that the asymmetry can change with the level of effort. However, some riders are willing to accept this inherent inaccuracy and imprecision.
Because neither the old Polar nor Ergomo power meters used strain gages, their accuracy and precision could not be statically checked in the field by the cyclist; they could only be checked dynamically (or against anotheher known-calibrated power meter).
The unreleased Garmin Metrigear and Brim Brothers pedal or pedal cleat power meters are rumored to use piezoelectric sensors and solid state accelerometers rather than foil strain gages but until they reach the market all claims about accuracy or precision should be taken with grains of salt. An interesting problem in the design of a pedal- or cleat-based power meter is that the direction of force and the position of the pedal spindle must be known: for example, if you add downward force at the bottom of the pedal stroke, that is wasted force since it does not aid in moving the crank in the correct direction; likewise, if you press down (however slightly) on the upstroke, that will cancel out some of the force exerted by the other leg on its downstroke. Keeping track of the various force vectors is therefore key to getting reliable accuracy and precision. To some extent, the Stage power meter can also occasionally be susceptible to a related problem: the Stages uses a solid-state accelerometer in the pedal (similar to the solid-state accelerometers one can find in smart phones) to determine its position. Early production models of the Stages were plagued by imprecise measurements of pedal position, so pedal speed was also imprecise — and this had repercussions for the precision of the final power estimates.
The recently released (as of January 2012) Look/Polar power meter uses strain gages arrayed along the pedal spindle, and each pedal must be carefully installed so the pedals know which direction the forces are being applied -- a special tool is supplied with the pedals to help with the orientation. To simplify the conversion of measured forces to torque values, the Look/Polar pedal allows the use of only four different crank lengths: 170mm, 172.5mm, 175mm, and 177.5mm. Cranks shorter than 170mm are currently not supported. One pedal is the "master" and the other is the "slave"; the slave pedal transmits information to the master which then bundles data from both pedals and forwards it to the head unit. At the moment, the Look/Polar pedal uses its own transmission protocol and no other manufacturer has yet signed on to provide compatible head units. Early reports on the new Look pedals confirm that the orientation of the pedals is critical: because the spindle of a pedal is small, a small absolute error in alignment can be a large relative error in its angular orientation.
The iBike takes a completely different approach: it calculates power indirectly. That is, you need a certain amount of power to overcome changes in potential energy (climbing or descending), for changes in kinetic energy (accelerating or decelerating), to overcome aerodynamic drag (including the wind) and drag from rolling resistance so if you know ground speed, gradient, wind speed, your total mass (you plus bike and all equipment) then combined with estimates of the coefficients of rolling resistance (Crr) and of aero drag and front surface area (CdA or drag area) you can calculate the overall power (see, for example, here). In essence, the other power meters on the market focus on the "supply side equation" by measuring power supplied by the rider somewhere along the drivetrain; the iBike focuses on the "demand side" by measuring power demanded to move the bike against wind, gradient, and other drag forces. Under normal conditions, this can be quite (perhaps even surprisingly) accurate, though the precision of the power estimated in this way is not as good -- the iBike assumes that the aerodynamic drag area (aka CdA) is constant, so if the rider changes position (say, moving from the drops to the bar tops) or if the wind speed differs because the yaw angle changes, the power estimate will be off. In general, the iBike has been shown to be quite accurate for hill climbs; less so for rolling courses or riding in a pack, so the overall accuracy will depend on the exact mix of riding done and the variability in the direction of the wind. As with the non-strain gage based old Polar and Ergomo, the iBike cannot be statically checked for accuracy or precision; worse, nor can it be checked on a dynamic rig in a lab since it depends on gradient and wind speed. Checks of the iBike have been performed in the field when riders have mounted another power meter on the same bike and compared the two data streams.
There have been a few "simultaneous" comparisons of power meter accuracy where one rider mounted two or more power meters on the bike and went on structured or unstructured rides. You can see one such "Rosetta Stone" comparison here and here.
In general, all of the commercially released power meters have been accurate (and sometimes precise) when newly-adjusted and performing under ideal conditions. However, conditions are not always ideal and parts get damaged, dirty, and deteriorate. If accuracy and precision are important then the "design" accuracy (whether based on strain gages, optical sensors, magnetic sensors, or wind speed sensors) is only half the battle: equally important is the ability to verify a power meter at home so you can tell when they're off.
I found this image of the first SRM power meter interesting:
The crank is rigged up like a lever (rotating around the spindle) - the more forcefully you push the pedal, the more the strain-guage bends, the output of which is used as part of calculating wattage (as is better described in the other answers!)
Many modern power-meters are essentially refinements on this concept
Fundamentally, power meters all work by measuring force (or torque) and a speed.
http://en.wikipedia.org/wiki/Power_(physics)#Mechanical_power
P(t) = F(t) * v(t)
In other words:
Power = Force * velocity
A pedal or crank based power meter will be measuring either how much torque is applied to the cranks. That combined with your cadence gives you the power output (add in time and you have the total work done).
A hub based power meter is measuring the torque applied to the hub by the drivetrain and combining that with speed or wheel rotation counts to do the same basic math and calculate the power output there.
Basically, one measures the power going into the drivetrain, the other measures power coming out of the drivetrain. If you had both, the hub power meter would show a slightly smaller amount of power due to power lost in the drivetrain. A bicycle drivetrain is very efficient however (especially when well maintained), so I wouldn't worry about the very small difference. To some extent it may depend whether you care more about your power output or the power output of the entire rider+bicycle system.
The actual underlying physical mechanism is probably a strain gauge consisting of thin wire running zig-zag over a beam that bends slightly as force is applied. Bending the wires changes the electrical resistance. http://en.wikipedia.org/wiki/Strain_gauge
I've also heard of power meters that measure force to the pedals. I assume they need to be told the length of the cranks. I've also heard of systems that measure chain vibration to calculate the tension on the chain to get force/torque information to calculate power output.
