It's not an "analog to digital convertor ". it's a Digital to Analog converter -> hence the name -> DAC .. or in other acronyms: D/A Converter .
a DAC, is a device which converts digital signals into analog signals.
Just think of your mp3 file, as a whole paragraph of notes with words with rhythm (and so on...), written in a strange language..
Now, you need a translator, which will translate the mp3 file, to the analog signals -> the language understandable by your Phaitons.
The better the translator (the DAC) is, the better the "source quality" is.
of course, if you hear 32kbps mp3 files. or you just hear music through "YouTube", there isn't much information to translate there.. so you can now say, that the "signal quality", or, "source quality".. is just low.
a DAC will only translate the file, to the signals (the "language"), later understandable by your Amp. (later, the headphones / speakers).
the Amp. is needed here, to boost the signals, so the headphones will be able to "hear" them loud enough, so you could hear them, too.
When you say that the sound from your iphone is "crappy", compared to the sound which comes from your desktop -> you're actually saying, that your desktop DAC/Amp device, is better than that of your iPhone's.
If you want to "upgrade" the iPhone's SQ (acronyms for: "Sound Quality") , you'll need a Portable Headphone Amplifier.
there are many, and usually, they are cheaper, compared to desktop Headphone Amps.
As always, in the audiophilia arena, there are always exceptions regarding the cost,
I think that the biggest comparison made (someone will eventually correct me if i'm wrong) , is in this link:
http://www.head-fi.org/forum/thread/214588/review-portable-amp-roundup-56-portable-amps-reviewed-and-compared-final-update-12-20-10-added-rsa-sr-71b
You are welcome to ask any question you might now have
-Eyal