Hi-fi headphones+headphone amplifier+portable player=portable hi-fi zone.

That’s about it, really.

The situation’s been improving, but usually portable players’ own headphone amplifiers are built to save energy, not for sound quality.

The problem here is that a headphone output of a soundcard, a portable player, a portable gaming console, notebook, etc. is usually not built to drive hi-fi headphones. It can also be rather coarse (remember that companies like Dell, Compaq, etc. are especially keen on cutting costs in mass production, saving on parts like sound logic) and fuzzy, and have the output signal suffer under a load.

One of the downsides of dynamic headphones is that they will perform as well as the amplifier that drives them allows. Electrostatic setups are usually sold with matching amplifiers (the STAX company makes a set of electrostatic earbuds, sufficiently cheap and very portable, by the way), but dynamic headphones have to be matched with what’s out there.

An amplifier provides enough stable current to drive headphones properly. Portable players’ headphone outputs are limited in power - a portable player needs some current for itself, and will likely “sag” under load if driving higher-impedance headphones, losing contours, texture, volume and “deflating” soundstage. This is especially true for “wonderful power-saving” players working on a single AA battery or two AAA batteries.

Headphones with an impedance of 600 ohms require ~8-12V of power; 60-ohm headphones may do with 1.5V, but will not go very loud.

Some headphone amplifiers are powerful enough to drive two pairs of headphones at a time with a splitter (as long as their nominal impedance is close, e. g. 55 and 60 ohm). Amplifiers built with a rechargeable “9V” brick battery (or two, or four batteries) have enough power for that, and also for driving higher-end headphones with an impedance of 300-600 ohms.

A common misconception is that an amplifier might make headphones deafeningly loud. In reality, the opposite is true - an amplifier provides enough detail for less loud listening. It is the lack of detail that makes listeners “turn up the volume”, as peak hearing sensitivity is at threshold of pain.

A properly-built headphone amplifier will also smooth out and improve a signal output by a soundcard or any other device; a Chu Moy amp improves the output of a Creative soundcard noticeably.

Earbuds, canalphones and very efficient low-impedance headphones may not benefit as much from the amplification as from the smoothing of signal provided by an amplifier. A smoother output signal causes less listener fatigue and lets the listener’s perception concentrate on music, instead of focusing on discerning music (this is an effect that’s not always conscious, but listening to an electronic sound source is all the more difficult the more artifacts there are).

Update

A headphone amp makes a difference. A really big difference in some cases.

An amp is especially noticeable with a digital audio device, like a soundcard. The clarity (and smoothness!) of source signal matters a lot, too.
The CMoy amp here, when combined with Foobar2000 in ASIO output mode and Secret Rabbit Code resampler to 96 KHz, makes all headphones sound natural enough, so that even the difference between AKG K-240 Studio and humble Kossart Portabel-Pro isn’t that apparent. Ironically, the less clear a source, the more fuzzy and coarse its output signal, the more it will make the differences between headphones stand out. The Kossart headphones have a smaller soundstage, a shallower dynamic range and a little dip in midrange, but the music output is lifelike enough for those differences to be barely noticeable.

For a digital source, a headphone amp acts as an additional smoothing filter, softening the original output and making it more pleasant and natural-sounding.

At the time of writing, Chu Moy amplifiers are sold on EBay for as low as $20-25, with $35 paying for a good setup, and audiophile-quality setups not exceeding $100.

As usual, it is better to listen once (to a headphone amplifier) than read about it many times.