Motherboards usually have just a basic Realtek audio chip. Higher end motherboards will have better components for audio and an onboard headphone amp. Not all soundcards have, but many now have headphone amps as well. Without a headphone amp it's hard to drive the headphones to any good volume, unless you have very sensitive headphones. Headphones can be low impedance around 32 ohms or as high as 600 ohms and they are harder to drive. Headphone amps are desired to give you that volume with 600 ohm headphones.
A DAC (digital to analogue converter) takes the digital signal from the computer over USB, converts it to analogue, and send it to your headphones through a headphone amp. The only difference between a sound card and onboard sound is nothing. Usually a sound card is better quality, but some motherboards have very good sound nowadays. The difference between these and a DAC is the soundcards are sontroled in Windows through software. This gives you virtual surround sound for headphones for gaming. The software will give you an EQ, etc... A DAC doesn't have any of this. A DAC jsut gives you an output device under sound in control panel and that's it. If you want surround sound through your headphones you need Razer software or a more expensive option that does exist. If you want an EQ you do it through your music player such as Foobar or Winamp. Also the DAC will be of much better quality sound wise than the soundcard. SOundcards just give you the software that isn't very valuable to begin with.
You can play 7.1 through headphones, but it's virtual surround sound. The sound is changed to simulate surround sound. Most motherboards and soundcards use Dolby headphone which is not rated too good. Some use creative onboard and creative soundcards have their own version called CMSS and it's rated better. And the free Razer surround is a download and it's rated very well. There are other professional tools as well but they are expensive. You can get gaming headsets with 3 speakers in each ear and get real surround sound. For gaming they are very good, but for everything else not so much.
And what a headset does is have a little box that plugs into USB. It's a cheap dac and headphone amp, and your headset plugs into that. And it runs with Dolby software usually, or Razer surround. Your better off buying your own DAC, own headphones, and getting real stuff.
I can tell you from experience that onboard audio hisses as low volume. A DAC is dead silent and just plain works awesome and sounds amazing. No software or driver issues to deal with like some soundcards. It just plain works, and works good. And owning a DAC means you don't need to buy expensive motherboards just to get better onboard sound. And having regular headphones means they last forever. No gaming headsets to break. It's a cheaper long term investment that actually gives better audio. And you can use the headphones for other purposes such as music, TV, movies, guitar, etc...
For DACs check out Emotiva. Their big and little ego are great. M-Audio also has the Super DAC that is great. Both can be hooked up to a home theatre stereo as well, not just headphones.