This shotgun effort is one that DeadEars wrote for a new Head-fier. The only small correction I'd make is that the "sound" of a DAC mostly comes from the output stage and not the DAC chip itself. Other than that, I think it's a very good effort.
OK, I'm bored and avoiding work, so let me give this a whack.
The hardest part about this for a newbie is that there are so many different possible routes to good sound. But for really high fidelity, you need to focus on three areas
1) source (recorded media) and playback method
2) amplification (highly dependent on 1 & 3)
3) sound transducer (what type of headphone, speaker, whatever is translating electrical signals into sound)
A few comments on source/playback
Your high quality source choices are perhaps the most critical. A few lucky people have access to dubs of master tapes, the actual first recording of the microphone signals when the sound of the original event is transformed into electrical signals. These used to be transferred electro-mechanically to vinyl phonograph records.
But today's most popular sources are probably CD's and MP3 downloads. The original analog sound signal is sampled at an extremely high rate, and the sampled values are saved as one-zero bits on digital media such as an optical disk or a computer's hard drive. This is called Analog-to-Digital conversion (or AD). The best quality is achieved when the original recording is done well, and at a sufficiently high sampling rate to preserve as much as possible of the original event. The lower the sampling rate, the more information that is "thrown away" and cannot ever be recovered during playback. Lower sampling rate is desirable because the sound data takes up less storage space. Most MP3 players have only a small amount of digital storage space, so if you want to carry a bunch of songs around, it is good if they take up very little space.
Higher sampling rates are better
But now that computer storage is so cheap, it makes less sense to throw anything away. Some sampling schemes use computer algorithms to decide which information to throw away, and try to keep the most acoustically significant information while throwing away empty bits, or those that "don't matter." Audiophiles mostly hate all of these "lossy" encoding programs, even though many people cannot hear their effects at higher sampling rates. True audiophiles want the highest practical sampling rates (CD standard is ~44.1Khz) and they want to keep all the bits, even if they contain little audio information. The standard lossless storage formats are WAV, Apple Lossless or FLAC, among others. A number of music providers are increasingly providing source data at higher sampling rates, for instance 88.1Khz which is twice the sampling rate of standard CD's, or even higher 96Khz/24 bit recordings. Once you have heard a great recording, using 96/24 sampling rate, you'll never want to listen to MP3's again!
Digital to Analog Conversion
As a high-fidelity enthusiast, the next issue is how you are going to turn the digital bits back into signals that will drive headphones. This is what the DAC -- digital-to-analog converter -- does. The DAC usually has two components. One that is the chip which does the decoding (and a bunch of support chips/components). Examples might be DAC chips from Cirrus, or Burr-Brown, or Texas Instruments. While they all do the same conversion, they approach the problem somewhat differently, with audible consequences. Picking the best DAC for you means listening to how they work relative to what sound profile best meets your ideals/needs. The above advice (go forth and listen) is the best way to pick a DAC and headphone amp!
The second part of the DAC is amplification of the tiny signal coming out of the DAC chip. Most DAC's use another chip called an Op-Amp to do this. Op-Amps can be powerful enough to drive headphones directly, but more typically, they will output a 1-3 volt signal. That signal is an audio signal (analog) which needs to be amplified to drive headphones or speakers.
So to your question about connections. If your source component is a CD player, it probably has a built-in DAC, and on the back it will have stereo output jacks (RCA jacks). The signal out of these is already converted to analog, but will need to be boosted to a higher voltage/current in order to drive headphones or speakers. On the other hand, if your source is a computer, you probably will need to use an outboard DAC which will convert a digital signal from the computer into an audio signal that you can listen to. The computer may have a built-in "sound card" which does D-to-A conversion, but typically the sound quality isn't all that great. So for best audio quality, you need to get the digital signal out of the computer, and hopefully do this in a way that is least harmful to the audio quality. One way that many audiophiles approach this problem is to use some form of USB (universal serial bus) connection. A DAC might have a USB input, so all you would need to do is get a USB cable and connect the DAC to the computer. Most computers these days will recognize the USB DAC as a legitimate output device and allow you to send your audio signals there rather than to the internal computer sound card. Thus the DAC will have an input side (such as USB, or Toslink, or digital coaxial, or AES/EBU or whatever). And the DAC will have an analog output side, which is likely to be RCA jacks (so-called unbalanced audio connectors).
All that work, just to get a weasily little 1-3 volt analog signal. Sigh. Well, speakers and headphones are transducers. They work by moving air. Pushing air around requires a certain amount of power. Usually a diaphram of thin-but-stiff material is attached to a magnet coil. When a current is applied, the coil or the diaphram moves, and the movement pushes air. The air pushed is proportionate to the signal. Thus music is magically recreated, sometimes years after it was first created. Magic!
The power to push comes from an audio amplifier. The amplifier takes the tiny signal coming out of the DAC and makes it into a bigger voltage/current signal. The input side of the amplifier has jacks to accept signals from the source (your CD player or your DAC). Typically these are RCA jacks (or XLR if using balanced connections). On the output side are connectors for the headphones. Typically these are 1/4" jacks for the large plugs on high-end headphones.
So you have several "boxes" in an audiophile system. Source component(s) like your computer + DAC (or your CD player), Amplifier, Headphones. You have a USB cable to connect your computer to your DAC. A pair of RCA cables (one for right channel, one for left) that connect the analog signal out of the DAC to the input side of the amplifier. And finally output jacks on the amplifier that let you plug in your headphones.
The actual technology within the amplifier doesn't matter very much. It can use a diabolical combination of transformers, tubes, transistors, op-amps, or piezo-electric thingamajigs. It really doesn't matter. What matters is how it sounds to you, with your equipment. The accumulated wisdom of Head-Fi should be a great help, to see what combinations of source, software, DAC's, amplifiers and headphones have worked for others. Of course, free advice is often worth what you pay for it. So use your own ears, not someone else's. Go forth & listen.
And above all, have fun!