What is Headphone Impedance?
Headphone Impedance is the electrical characteristic of the headphone voice coil and magnetic field coupling of the voice coil & magnet inside the headphone.
Impedance is an electrical unit which expresses the combined Resistance, Inductance and Capacitance of the headphone's voice coil.
In terms of matching to an amplifier, the headphone impedance is the "Load Impedance".
An amplifier drives a headphone with voltage, the current drawn by the headphone is proportional to the headphone's impedance. 
As a rule of thumb, the load impedance (headphone) should be at least eight times higher than the amplifier output impedance.
This allows the amplifier to exert more control over the headphone, i.e. the lower the output impedance of the amp, the higher the damping factor and the higher the efficiency of the amplifier/headphone interface (See note 1 for additional detail).
Some basic electrical theory:
  1.  Voltage = Current x Impedance,      in AC circuits, i.e. music, power lines, etc.
  2. short form for voltage is "V" and is expressed in Volts
  3. short form for current is "I" and is expressed in Amps 
  4. Power =  Voltage x Current,
  5. short form for power is "P" and is expressed as "Watts"
  6. Impedance = Voltage/Current,       short form for impedance is "Z"
  7. Resistance = Voltage/Current,       short form for resistance is "R", this is for DC circuits, i.e. power supplies, batteries
  8. see note 2 for definition of Voltage and current, see note 3 for definition of resistance, inductance and capacitance.
Here's an example:
For a headphone to receive 2 watts of electrical power (this is almost never the case, just a simple example) it can be driven by:
  • 1 Amp @ 2 Volts , or
  • 2 Amps @ 1 Volt.
therefore if 1 Volt draws 2 Amps of current the headphone has an impedance of:
Impedance = Voltage/Current (or Z = V/I)
therefore Z =1/2 Ohm. 
Based on Z=V/I, we can deduce that Case 1 will have an impedance of 2 Ohms, and Case 2 will have an impedance of 0.5 Ohms (see note 4).
Again, this is just an example, headphone impedances are actually much higher.
Headphone power levels are almost always much lower, for example: 1 - 10 milliWatts.
1 milliWatt (mW) is 1/1000 of a Watt.
If a 600 Ohm headphone draws 10 mW, Voltage = 2.44 V, current = 4.1 milliAmps (mA)
If a 32 Ohm headphone draws 10 mW, Voltage = 0.56 V, current = 18 mA
Pros and cons of high impedance headphones:
  • Headphone impedance is usually increased by thinner wire and most importantly more turns of wire in the voice coil. More turns or loops creates a larger field (area of magnetic influence). In layman's terms more magnetic force for the coil to move the diaphragm.  Thinner wire usually works out to a lighter, more responsive diaphragm. Depending on the headphone design, this may lead to more accurate response.
  • The displacement (amount of movement) of the diaphragm (the part that vibrates to produce sound) can be better controlled via a more accurate flux (magnetic field to pull and push the diaphragm).
  • Difficult to drive for small headphone amps with low output voltage and low gain.
  • Most high impedance headphones need an amplifier with higher voltage gain and higher output voltage, e.g. the 600 ohm Beyer DT770/880/990 series. 
  • Allow Solid State Op Amps to work more efficiently with less distortion. Have a look at Op Amp data sheets and a graph of distortion vs. output impedance for most audio Op-Amps and you'll get the idea. This is a very complex subject, but most Op Amps are designed to output (typically) up to 10 Volts into 600 ohm loads or higher.
Pros and Cons of low impedance headphones:
  •  Headphone impedance is usually decreased by thicker wire and less turns of wire in the voice coil. The magnetic field is built up by more current.
  • Easier for small and/or portable headphone amps to drive. For example: an iPod or MP3 player headphone jack. Many small and/or portable headphone amplifiers are designed to output a volt or two into low impedance, high efficiency headphones, e.g. Grado headphones.
  • Low impedance, low efficiency headphones usually sound better when driven by a desktop amplifier, e.g. Audeze LCD-2 or AKG K70X.
  • Low impedance headphones usually sound better when driven by a solid state or a transformer coupled vacuum tube amplifier.  Low impedance headphone do not usually work well with Output Transformerless vacuum tube amps.
Generally, a properly designed desktop headphone amplifer can drive high and low impedance headphones and can drive headphones of any efficiency.
Do high impedance headphones sound better than low impedance ones?
No, there are many very good low impedance headphones available and many very good high impedance headphones available.
In headphones, the sound also depends upon the following:
  • Frequency response, this is a representation of volume decreasing or increasing with frequency, this is actually the magnetic field pulling and pushing with different amount of force at different frequencies, impedance affects this in complex ways.
  • Distortion, represents the amount of "change" from the actual signal to the real signal (i.e. input signal to output signal).
  • Build and design, i.e. sealed vs. open, etc.
  • Diaphragm design, voice coil design
To conclude:
High impedance headphone:  requires more voltage but less current
Low impedance headphone:   requires less voltage but more current
Amplfier output impedance: the lower the better for any impedance headphone
higher efficiency (for example:  102 dB/1 mW) less power required
​lower efficiency: (for example: 91 dB/mW) more power required
How they effect sound is dependent on the magnetic field generated and diaphragm construction (physical response of diaphragm).
We hope this clears the air.
A few notes:
  • Some people prefer the sound of vacuum tube headphone amplifiers which usually have higher output impedance relative to solid state (transistor)  amplifiers, and therefore have poorer damping factor and hence, fuller bass.
  • Voltage is electrical pressure, current is flow of electrons
  • Resistance is constant with frequency, capacitance creates lower impedance at higher frequencies, inductance creates higher impedance at higher frequencies
  • Headphone impedance is actually quite complex and different driver designs have different impedances at different frequencies and (to a lesser extent) voltage amplitude
  • Assumes diaphragm construction to be similar and magnetic field to be similar
  • Assumes linear response at all frequencies (not possible for dynamic driver headphones)
  • Assumes that impedance increases with change in coil characteristics only.
  • Assumes speed of electrons in an electrical circuit is constant, it does not change with voltage or current or power.
  • Other headphone types which do not have a voice coil (for example, those sold by Hi Fi Man, Audeze and Stax) can also characterized by their electrical impedance.