Unlike consumer two-channel stereo, there is no standard for expressing power output for a headphone amplifier. Most manufacturers of portable equipment usually just state power as nn milliwatts @ a specific impedance (usually 32 ohms). As portables have gotten smaller, they are being designed to run on 1.5 volts instead of 3 or 6. Power specs have shrunk as well. The smallest MP3 players today are usually capable of just 10 or 15 mw of power from the headphone jack. The larger and more robust units are capable of 30mw. For comparison, my Denon home CD player claims 70mw @ 32 ohms from the integral headphone output.
How much power is really needed for headphone listening depends on several things such as the sensitivity of your headphones, (expressed in decibels per mw), and how loud you like to listen to your music. If you push a player beyond it's available power it will start to clip the audio signal. Brief clipping of the audio is not usually audible, but ongoing clipping is very apparent as the sound becomes harsh and distorted. But usually, if your portable does not have enough power, you won't be able to get satisfactory listening level even with the volume turned all the way up.
Acoustic levels are measured in decibels. This is a logarithmic scale because the sensitivity of the human ear is not linear in the way it perceives sound levels. For a sound to be perceived as being twice as loud as another sound, it must be 10 decibels louder. That 10 decibels is really ten times the acoustic energy of the softer sound, and requires the equivalent of 10 times the amplifier power expressed in watts or in the case of headphones milliwatts to reproduce (assuming 100 percent efficiency in converting electrical power into acoustic energy) by means of an audio transducer (the headphone voice coil and diaphram).
The smallest difference in acoustic output between sound levels that is usually audible is 3 decibels, or 3db. Although perceived as being just slightly louder, this 3db increase actually requires twice as much power from an amplifier. Raise the volume another 3db and you have again doubled the power required from the amplifier (a total of 4x the power from where we started.) You can see how raising the volume on your headphones even a small amount has a big effect on the power demand from your amplifier.
Music varies in level from just barely audible to almost painfully loud (usually for just briefest of musical peaks). The difference between the peak and average level depends on the type of music. The dynamic range of most rock, techno, and pop music is highly compressed so the difference between peak and average levels are in the range of just 6db. For other music, the difference can be 10db or even as much as 14 db.
If we use 10db of desired dynamic headroom as an example, an amplifier that can produce a maximum of 30 mw of power can deliver 3mw for average music levels with enough dynamic headroom to be able to reproduce musical peaks that are 10db louder without clipping the signal. If we raise the volume just 3db, the average power demand is now 6 mw and the peak power required would be 60mw. As the amplifier is capable of just 30 mw, the amplifier will likely be clipping the loudest peaks, however, many amplifiers can sustain peak output levels that are 2 or 3db higher than the rated power for very brief periods (measured in milliseconds), so the output will not be clipping on these brief music peaks. If we continue to raise the volume, again by 3db, the average power level will rise to 9mw and the peak power demand will be 90mw (almost 1/10 of 1 watt), and the output of the amplifier will be clipping on the peaks of the music.
All dynamic headphones, with one notable exception, require far less than 1 watt of power to reach their maximum output. Many do so with far less than 1/10th of 1 watt of power. A headphone such as the Sennheiser HD-580 (also 600 and 650) that requires 200~300mw for full output can still be driven to very loud levels with an amplifier capable of 70 or 100mw, but will likely not be able to be driven to high enough output from an amplifier capable of just 30mw without audible clipping. The AKG K1000 headphones, with the lowest sensitivity of any dynamic headphone, require approximately 100X the power of typical headphones to achieve realistic output levels. Amplifiers capable of driving these phones need to be able to deliver between 3 and 10 watts @ 120 ohms.
For a given power level, if your raise the impedance of the headphone ( or speaker), an amplifier must be able to deliver a wider range of peak to peak (+ to -) voltage through the cycle of the audio signal. At some point the amplifier is no longer able to deliver the needed peak to peak voltage and the amplifier will clip the signal.
Portable amplifiers are typically challenged driving high impedance phones, not because of the sensitivity of the phones, but because the portable amplifiers are limited by their supply voltage (usually just 1.5 or 3 volts). By comparison, most home solid state headphone amplifiers have a DC power supply that provides a constant 12 or 14 volts. Some home/portable headphone amplifiers use a 9 volt battery to power the amplifier, and a few even use (2) 9 volt batteries (one for the positive side of the power supply and one for the negative side ofthe power supply), for 18 volts.
Amplifiers have output impedances that range from very low (almost 0 ohms) to very high (several thousand ohms, as with some tube amplifiers). For efficient transfer or power, and to avoid potential frequency response anomalies, the output impedance of the amplifier should be many times lower than the impedance of the headphones. Tube amplifiers generally solve this by using output transformers. Sometimes, with high impedance headphones, the output of a tube amplifier can be sent directly to the headphones without the use of a transformer. This is inefficient, as no more than 5 or 10 % of the potential amplifier power is available to the headphones but this may still be enough for achieving full audio output levels.
Older solid state amplifiers and receivers generally use the main power amplifier to drive the headphone output, using level matching resistors of between 100 and 300 ohms to more closely match the level of the music when listening to speakers to the level of the music when headphones are used. This avoids the abrupt change (an increase) in volume for the unwary user that would otherwise result. At the time this equipment was originally sold the better known premium headphones from brands such as Sennheiser, AKG, Koss, and Yamaha were high impedance phones (150 ~600 ohms). These older amplifiers and receivers continue to work well with today's high impedance phones, but with low impedance phones such as Grado (32 ohms) there can be a noticable rolloff of the highest frequencies due to the impedance mismatch. At least one owner has stated that they like the way their Grado phones sound when connected to one of these older receivers, but most Grado fans would find the high frequency response rolloff to be unacceptable.
