Rather than focus on terminology, I think the OP just wants to know how audio quality can improve in any sense.
King-Reyes, focusing on just the headphones, you must first understand that the typical budget headphone really mangles audio. Many headphone drivers (the part of the headphone that produces the physical sound) are not responsive enough to correctly follow subtle details, and many do not produce all frequencies equally but instead have great difficulty (both in volume and in detail) with high and low sounds. Due to resonance with their enclosures and cheap engineering in general, they'll have severe spikes and dips in their frequency response and also be simply unable to follow an audio signal with tight accuracy and control.
A good headphone will have much better frequency balance - with no exaggerated spikes or dips in random places - and much better detail, being more closely responsive to the signal. This will translate into much more detailed sound (so much more that most people hear entirely new instruments and parts of songs they though they knew well!), much richer and smoother sound (due to more even frequency response), and overall greater realism, presence, beauty, and texture.
The actual mechanical ways that this is accomplished are better materials, resonance-free/canceling enclosures, methods of attenuating or letting pass certain frequencies, closer matching of left and right drivers, higher quality manufacturing methods, higher grade electronic components, and above all a much better designed and built driver model.
However, there is a whole lot more to head-fi than just headphones. I'll just consider digital in the following:
First, there's the original data (for instance, on your CD). On a good quality song, this will have been mixed and mastered by a professional recording company that knows the technical aspects of the process well. Audiophiles like to use exactly the original source data, or else a format that shrinks the filesize without losing any of the information. This is called lossless encoding, and preserves 100% of the quality. Examples of lossless are: CD, wav, FLAC, Apple Lossless.
If most of your music is in 128 kbps MP3 files, you may want to start re-ripping your music in something better quality. MP3, along with AAC and Ogg, are examples of lossy encoding, i.e. ways of shrinking the filesize that LOSE some of the original info. These formats
can be pretty good, but it might be a waste of money to spend on expensive headphones and amplifiers unless you are using a higher bitrate. For instance, many people are fine with 192 kbps AAC files, which are more difficult to tell from CD quality than 128 kbps MP3.
Next you need some way to play your digital file. This can be a high-quality CD player, a computer with a very good soundcard, or a computer connected via digital cable (optical, digital coax, or USB) to a standalone DAC (digital-analog converter). There are other possibilities, but with CD audio these are some of the most common. An iPod is not the best source, but it's fine for portable use.
What all of these have in common is a DAC (digital-analog converter). In a high-quality CD player, a good soundcard, or a standalone DAC, this component will be of good quality. Quality here means that the DAC produces an analog signal that is close to what the original data represented. In most consumer-grade equipment - such as cheap soundcards, laptops, portable CD players, etc. - the very cheap DAC does a surprisingly poor job of producing an accurate analog signal. Therefore many audiophiles who use digital sources like to get an amplifier with a built-in DAC.
Which brings me to the amplifier. The point of an amp isn't to make headphones louder, it's to deliver enough current to drive the headphones correctly but without messing up the signal in the meantime. Again, cheap amps built into portable devices such as the iPod or computer headphone jacks are usually not up to the task of delivering enough current for upper-level headphones. Not to mention, they tend to introduce noise and distortion or to inaccurately amplify the audio with respect to frequency balance. A good quality amp will very cleanly and accurately amplify the signal to a level that can drive even the most power-hungry headphones well.
So, to summarize, quality means detail, accuracy, richness, tonality, etc., all of which will result in more realistic audio, more things to hear in your music, and just plain better sounding results. Quality is attained through the following chain:
Data: lossless or high-bitrate lossy
Source: anything with a good DAC inside, that will create a nice starting signal
Amplifier: anything that will cleanly and powerfully increase the current of that signal
Headphones: anything that will accurately convert that signal into physical sound
It's amazing how in most consumer setups each one of these stages is rather poor quality. A lot of improvement can usually be heard with an upgrade as low as $50! After that the sky's the limit for spending.
Good luck,
GLL