The output device
We're going to assume you want to jump off the motherboard and go directly to an external device. Depending on what you want your speaker setup to look like, how much room you have, and your budget, we have a few different options.
There's a school of thought that says relying on your motherboard's output introduces interference and noise, caused by all the other components. This is true, but on some motherboards, manufacturers have gone to decent lengths to minimize interference. It's still there, though, as other components add line noise, since everything's connected to the same power supply, and there's guess work in that component, too. Our advice is to skip the PC box altogether and get an external DAC that will run over USB.
A good USB DAC will give you several important features. One, you're likely going to get a good DAC chip (AK, ESS, and AD chips come to mind). Two, it'll have a good built-in headphone amp to drive high-end headphones. And three, you can use many of them with your laptop. All that aside, the audio components and path design are far superior to than anything you can get inside a PC.
The budget option (less than $100)
You can go with small USB adapters for audio, and some of them even have 5.1 channel output. Most of these range between $6 and $9—yeah, really—and offer you working audio output. These are no-frills, no high-resolution audio, and some of them sound even worse than standard onboard audio. We don't recommend going this low, unless your onboard audio is shot.
Creative Labs has its Sound Blaster Omni, which is a USB sound module with built-in volume control. The Omni is small enough that you can tote it around with you if you're traveling, and it's entirely powered by USB; there's no power brick to haul. It offers relatively good sound.
If you want to go even smaller, M-Audio has its new Micro DAC that can do 24-bit/192kHz output. If you want a good step up, this is what we'd go with, but unfortunately it doesn't offer 5.1 output over analog. So for gaming, you're better off with the Omni as it has 5.1 analog speaker output and will do 24-bit/96kHz output. If you just want stereo, Schiit's Modi 2 supports 24-bit/192kHz sampling rates if you're willing to install drivers. Schiit's Modi 2 Uber for $149 is a step above if you're looking for full 24-bit/192kHz output over raw USB and SPDIF, and it seems to be a popular option with several Maximum PC readers.
A step up:
You may want to consider an external USB DAC, but most within this budget only offer stereo output. Good options include the Schiit Modi 2 ($99), Optoma's NuFroce uDAC3 ($98), and FiiO's E10K ($119). They offer clean and low-noise sound output, but the caveat is that they're only stereo. If you're looking for 5.1 output, the Sound Blaster X-Fi Surround 5.1 Pro is a recommended choice.
The midrange option ($200 to $600)
This is where a lot more options open up. Many prefer to buy a receiver to handle audio duties. Even receivers in the $200 range can offer significantly better audio quality than any onboard solution. Going with a receiver will ensure that you have all the most popular sound formats covered, and in terms of flexibility, a receiver will let you use your desktop speakers or upgrade to hi-fi speakers.
Receivers also provide significantly more power output, and allow you to connect more devices. You'll be giving yourself better components, cleaner power delivery, and be covered for both stereo and multi-channel output—even budget receivers now have 7.1 channel support. The components inside a receiver are also better quality than can ever be stuffed onto the limited space of a motherboard.
The main drawbacks are that receivers are big and consume more power. If you're going to use a receiver for your PC, you're best going with an upgraded speaker system. You can still connect multi-channel desktop speakers to a receiver through its pre-outs, though.
Some good options include Onkyo's TX-NR636 8.1 channel receiver, which fully supports Dolby Atmos, should you choose to go all-in on speakers. Another good option is Denon's AVR-S900W, which is a 7.2 channel receiver; it lacks Dolby Atmos and has been discontinued by Denon, but it can still be found. You'll need to forgo your desktop speakers with receivers below $600, though, as they don't usually have the pre-outs necessary for use with desktop PC speaker systems.
For gamers, a good option in this price range is the Sound Blaster X7. It's got everything you need for excellent multi-channel and stereo output: a built in amplifier for passive speakers (in stereo), powerful headphone amplification, excellent DAC performance, and features for both music and gaming. It's even liked by the audiophile community and we picked it as the go-to option in our Dream Machine 2015.
The high-end option ($1000+)
If you have a bigger budget to work with, you'll have a lot more options still. This is where some serious DACs come into play. At this point you might even want two DACs, one for just listening to music and general audio, and a 5.1 unit like the X-Fi for games. All the high-end USB DACs are stereo-only.
For gaming, you'll definitely want a multi-channel receiver. If you do a lot of movie watching on your PC, this route will also serve you well. Some people use actual LED TVs for their PC setup—4K panels included. If this is you, a receiver is a must for 5.1 and higher. Any of the receiver brands mentioned above will have good options, but now that you're in this price bracket, you can consider other high-end brands like Cambridge, Marantz, NAD, and Pioneer.
For us, we're skipping the receiver entirely, focusing our money mainly on a high-end USB DAC that can work well with both speakers, monitors and headphones.