The first upgrade suggested to anyone with a new pair of headphones is usually a DAC. It is often the wrong one, or at least the less important half of the right one. The reasoning is worth walking through, because it also explains when a DAC genuinely does help.
What a DAC does
A digital-to-analogue converter turns the stream of numbers in a file into a voltage that varies over time. Doing this well was a hard engineering problem in 1990. It stopped being a hard engineering problem some time ago. Competent converter chips are inexpensive and are now built into phones, laptops, monitors and the dongle that came in the box.
That does not make every implementation equal. A converter sitting next to a noisy switching supply inside a laptop can pick up interference, and some built-in outputs have audible hiss or a high output impedance that interacts badly with low-impedance in-ears. Those are real problems and an external DAC solves them. But they are implementation problems, not conversion problems.
What an amplifier does
An amplifier takes that small voltage and makes it big enough to move a driver. This is where the actual physical constraint lives. A 300 ohm headphone needs several volts of swing to reach a normal listening level, and a laptop headphone socket may only manage one. When it runs out, the peaks clip. The music sounds flat, thin and oddly lifeless at higher volumes, and turning it up makes it worse.
This is the failure people most often mistake for a bad DAC. Nothing about the conversion has gone wrong. The output stage has run out of headroom.
When each one is worth buying
Buy an amplifier when the headphone is high impedance, low sensitivity, or both, and you find yourself near maximum volume on a phone or laptop. That is the case where the difference is not subtle and is not psychological.
Buy a DAC when your current output has an audible problem, hiss, buzz that changes when the laptop works harder, or interference that tracks the screen. Buy one when you need an input your source does not have, such as optical from a television. And buy one when it comes in the same box as the amplifier you needed anyway, which is why most desktop units are both.
The honest summary
If you have efficient in-ears and a modern phone, a separate DAC is unlikely to change what you hear. If you have a 250 or 300 ohm open back plugged into a laptop, you are not hearing what the headphone can do, and the part that fixes it is the amplifier.
We are not going to publish measurements to back that up, because doing it properly needs a test bench and a report we can show you. What we will do is print the impedance and sensitivity of every headphone we sell and the output power of every amplifier, so you can work out the combination yourself.