A sound card (or "audio interface") provides all of the hardware required for audio output or audio handling from a computer - it includes the DtoAs (DACs), AtoDs (to take audio in), and may include hardware acceleration for various features (in the form of specialized DSPs) - depending on what you need to do, you can get incredibly fancy with these (when you get into recording/pro interfaces that can handle hundreds of inputs at once, for example). The USB devices being discussed are the same exact thing - they are audio interfaces that provide all of the hardware required for audio output from a computer. They usually include amplifiers as well (which can give them a big edge over internal cards). They usually do not handle advanced processing or decoding features, and usually do not handle multi-channel audio; this may or may not matter to you (for the vast majority of users it's not an issue either way).
Standalone DACs will only accept a digital audio signal (PCM, DSD, and a few exotic formats from Accuphase, Esoteric, etc), and convert that to an analog audio signal. These devices are different from the USB devices discussed in this thread and the above described interfaces (they are not audio interfaces and have no knowledge of computers or anything like that). A DAC is a sub-set of any audio interface, regardless of how it plugs into the computer; USB, PCI, PCIe, FireWire, telepathy, etc.
The easiest upgrade path is a USB device, especially if you have a laptop, as it will include a headphone amplifier as well (this amplifier will make more of a difference in the grand scheme of things). There are numerous options at a variety of price points here. Modern buzzword compliant devices will tout asynchronous hi-speed connections among other things.
If you have a desktop system, a sound card is probably the most versatile upgrade path (RFI/EMI ghosts be damned), as it provides a more diverse and capable processing suite in one centralized device (but, increasingly, and as a result of changes in software development paradigms - this is all being offloaded to software anyways, so it isn't as big of a deal now as it was even a few years ago). You would then want to connect this new toy to an amplifier or a standalone DAC and then to an amplifier. Some sound cards, just like their external counterparts, include built-in amplifiers which are quite competent (the only real limitation being the 12V DC supply afforded by the computer). Unless you're connecting something absurdly hard to drive, the built-in amplifier boards should be suitable (you can probably quite honestly get away with a conventional card with the headphones you have).
As far as handling movies, music, games, and so on - years ago having a special sound card that supported HW3D audio was all the rage. That's done and gone now, for the most part. You can still buy fancy cards, but they are going away, in favor of more "audiophile friendly" stereo only devices (this is probably the only downside, is the lack of support for multi-channel audio aside from S/PDIF (which is inherently lossy)) - the USB solutions proposed above will work equally well (if not better, as a result of their included amplifiers (which can end up being exceptionally powerful, as they are only limited by the design's power supply)).
As far as "what sounds absolutely the best" - the components often used in outboard devices are no more blessed or special than what you'll find internally. That said, power supply limitations can become a problem (this is not going to be an issue in your case), and depending on your application demands you may find better features or better features at your price point in a certain solution. There's no real reason to preference one or another as if it were a paradigm. Get what you can afford and what suits your needs.
Regarding speakers - no you cannot conventionally hook them up to a standalone DAC; I know of a few devices that include pre-amplification abilities and can drive active speakers (speakers that include their own amplifiers); the Emotiva XDA-1 is an example. CEC makes a few devices as well; so does Grace. Speakers require much more amplification than do headphones - either invest in active monitors and select a headphone amplifier/driver that can function as a preamp as well, or just skip all of this and buy a stereo receiver or integrated amplifier (like the Harman Kardon HK3390 or Onkyo A-9555) - that'll drive your headphones, speakers, and give you other functionality (phono preamps become available, some have internal DACs, etc).
Finally, price is not a good measure of the quality or ability of an amplifier - more expensive does not imply "low impedance drive ability" any more than it implies quality or the device's use off-label to cure cancer. Impedance and sensitivity must be considered together - and be wary of headphones that have their sensitivity stated in dB/V (the K701's are spec'd this way; when you tear the specs apart you realize they're not incredibly efficient and all of the "hard to drive" comments make lots of sense). Low impedance and high sensitivity is fairly common for headphones - in this scenario you can essentially plug them into anything without concern. It becomes problematic for sources when the load is high impedance and low sensitivity (orthodynamics, for example) or has large impedance fluctuations across the frequency range resulting in the amplifier acting as a filter due to its inability to cope (HD 600's out of very cheap opamps for example). There does become a point where high end amplifiers have almost nothing in return for their price; do keep that in mind (this is relative to what they're being tasked with driving and what features they need to provide - there is no clear-cut bar). Also keep in mind that there are plenty of silly expensive devices out there that are entirely outclassed by things costing a fraction of the price; the devil is always in the details.
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Hmm good point.
But I'm still wondering. Maybe I should start from the basics. What does a sound card do? I understand the DAC based on what I've read, it converts digital to analog. A soundcard, from what I know, is just the audio processor for all things, including music. I'm wondering if it'll be a better option for me to get a bomb soundcard (maybe one with a built in DAC if there is such a thing) or if I should just deal with the onboard and focus purely on music and get a quality DAC. I'm not sure I have money for both.
And how does a DAC handle sound that's not music, like gaming and stuff? Or movies. Not that it's a big concern but it seems anything outside the music is the soundcard issue.
