I was confused at the beginning with all the options and recommendations available. I researched for a long time to find a good cost effective amp and dac, because I had never heard a good one before.
From what I have read, a good solid state amp with a low output impedance is your best bet. Tube amps have warm up time, and can change the sound(which some people like) solid states are also way cheaper and more power efficient.
A sound card is usually an amp/dac often with gaming features and software.
And for a dac, (digital to analogue converter) usually something external from the computer is better, to avoid electrical interference.
In my case, my laptop had a hissing electrical noise out of the headphone jack. So I bought a creative sound blaster omni, a USB sound card to solve the problem. I also wanted the other features, like sbx headphone surround, and the eq. It also has a headphone amp and a pretty decent dac built in.
For a while that was good enough. Then I decided to upgrade again just to see if it would make a difference.
I bought the schitt Magni 2 uber(headphone amp), and modi2 uber(external dac).
Of those two the Magni made the biggest difference. The sound became more detailed, better separation of details, and more impactful, especially in the bass.
The modi... Well there was a difference, but it took a lot of concentration to hear it. The main difference between the two dacs is that the omni was powered by USB, and the modi2 uber uses a plug in. The omni would occasionally have lesser quality because of unsteady USB power, especially when the laptop was running on battery power.
So now my setup is: omni>(digital optical out)modi>magni.
I get the universal eq, and headphone surround, optical out to the modi, which sends a nice clean signal to the Magni.
The two most recommended bang for buck discrete dac amp combos are the schitt modi and Magni, and the odac and o2. But you don't really even need to spend that much. I got a pretty good performance boost just from the $70 omni.
Unless you have some glaring issue like electrical noise, or a noisy signal, you could probably skip getting a dac as long as the one in your computer is good enough, (most aren't for me).
A good amp with low output impedance, is your best bet. It just insures that your headphones get all the clean, even power they need to perform their best.
I bought all of these products from amazon, my logic was that if I didn't hear an improvement I would just return them (30days).
All of my headphones sound better on the Magni, but it depends on the headphone. For some it makes almost no difference like the m40x it can run almost the same on anything. But for the he400, and hd558, the improvement is noticeable.