A DAC or Digital to Analog Converter takes the raw ones and zeroes stored on say a CD in bit form and translates them into an analog signal that your speakers / headphones use. In general DC voltage is bad for a speaker. A standard headphone or speaker accepts this analog signal and is able to produce a sound wave by moving a diaphragm or speaker cone in and out. This is typically done with some sort of magnet.
See general DAC wiki here:
http://en.wikipedia.org/wiki/Digital-to-analog_converter
For me a DAC not only converts the ones and zeroes, but it controls the noise in my signal path. A bad DAC will allow noise to latch onto the signal and get mixed in with your music. This is easily heard with most PC listening stations. Have you heard the mouse move or the little bleep-bloops of your processor? If so the DAC is most likely the culprit. That and how the power is implimented in your PC. A good DAC should be isolated from noisy power sources, convert those ones and zeroes without any phase shift or attenuation.
DC current usually harms the speaker because it moves the diaphragm / cone to a known maximum and leaves it on. Unless the DC current is switching on and off in some fashion the speaker will be stick in a flexed state. The AC current can finely control the speaker by controlling the on/off nature of the signal going to it.
Typically a DAC should not interfere with the signal path and thus should not change the sound. If you are hearing differences in your music listening to the headphone out on a DAC the most likely cause is what is being used to deliver the AC to your headphones / speaker. There probably is some sort of amp in there to boost the signal to a level that can actually be audible. Other factors include impedance which can affect the signal as well.
See transmission lines:
http://en.wikipedia.org/wiki/Transmission_line
Basically if the impedance is not just right either the driver will not be controlled well ( damping ) or there will be attenuation or worse reflection of the signal. This will be heard as noise or a decrease in volume.
So if you are listening to some music and then listen to it through a DAC and notice the detail missing, your are probably limited by the output of the DAC. Just remember that all a DAC should do is accurately and swiftly convert the digital signal ( ones and zeroes ) to an analog one for your speaker / headphones.
For me the biggest improvements to my listening experience are:
1.) Source material
You have to start will well-recorded and high quality music to have a chance at keeping the signal that way when it reaches a speaker. Loss, compression, and the original mastering are just a few of the variables.
2.) Headphones / Speakers
Once I have some "good" recordings I really notice an improvement in the sound when I change the speakes / headphones. This should be of no surprise as the coil, damping, size of the driver, and enclosure are all what actually deliver the sound to your ear.
3.) The last thing that helped was using an amp that had a low enough output impedance to properly damp the driver as well as enough voltage / current to deliver the sound I wanted. Many amps also have noise issues so a good amp will not introduce any more noise to the signal.
For more on damping a dynamic driver go here:
http://en.wikipedia.org/wiki/Headphones
To fully appreciate and understand how to get the most out of your listening experience I suggest trying a few pairs of headphones and a few amps. Find some good source material and play it through the amps paying close attention to how they sound with each headphone. In the end choose the combination you liked the most. Be sure to have reasons why you like it and accept positive criticism.