Before reading my rant, first review the video in question from Marques Brownlee, considered the "best technology reviewer on the planet right now" in 2013.
His video is about the quality of onboard DAC/amplifiers on the sound card embedded with most motherboards inside of computers. Firstly, he refers to it as "Digital to Audio Converter". Just google it - that does not instil confidence. Secondly, he makes this very broad statement:
As a general rule people buy external Digital to Audio [sic] Converters because the internal ones that are built in to our computers that feed directly to the output headphones jack basically, are typically really really bad.
The question is, did he perform a review of a statistically significant amount of popular motherboards before arriving at that conclusion, or is this tongue in the cheek talk? The problem is that his whole argument, the whole point of the video, is based on this one assertion. He makes this one statement without validating it. 20 000 people liked his video, and 1.18 million people watched it. If he made a false statement based on "feels" and not "facts", that is like being a bad parent to his 5 million subscribers.
His video was published in 2012. So we have to look at the state of onboard DACs at that time. Remember, he specifically stated that the onboard DAC is bad. He did not mention the amplifier or input/output impedance.
Read this article from Tom's Hardware dating back to 2014. They compared a standard $2 ALC889 DAC found on many motherboards with a $2000 dedicated DAC and found that:
Using world-class headphones, a $2 Realtek integrated audio codec could not be reliably distinguished from the $2000 Benchmark DAC2 HGC in a four-device round-up. Again, all four devices sounded great.
Also review this post by the guy who built his own (great selling) DAC. He has lots of technical information on DACs that is well worth reading:
IS AN AMP OR DAC NEEDED? The short answer is: "probably less than people think". A lot of people have no real complaints with the sound they're getting but they read others claims of huge improvement in the sound by adding often fairly expensive hardware. But sometimes the perceived improvement is mostly psychological--a well documented phenomena much like how a more expensive wine often tastes better than a much cheaper wine until you put them both in plain brown paper bags
One thing that is true, is that if you try to use audiophile quality headphones (which usually do sound vastly better than cheap consumer or gaming headphones) with the onboard DAC on a PC or Mac, you will probably find that it does not have the same punch as when using it on your Hi-Fi system. This is because computer amplifiers usually are rated for lower impedance headphones, typically 16 - 32Ω. Top end audiophile headphones can have an impedance rating of up to 600Ω. This would simply cause the volume to be too low to enjoy the full dynamic range of the music. In those instances, an external headphone amp might help. However amplification and impedance has nothing to do with a DAC. Converting a digital signal to analogue is one part of the process, the next part - very often handled by a separate chip on the motherboard - is an op-amp that amplifies the output from the DAC to a level that headphones / speakers can make use of.
Conclusion: Onboard DACs are just fine most of the time. Unless you hear specific issues such as popping noises, white noise, hum, clipping etc. rather invest that money in better headphones.