Define “regular headphone?” And how would you describe the sound of the onboard audio you have now, it’s pros and cons?
Usually, onboard audio on desktops and especially laptops is cold, hazy/muddy, and weak (especially for high impedance or low sensitivity headphones). If a “normal” headphone to you is a $50 or less portable headphone, like a Koss Porta Pro or Sennheiser PX-100, then it wouldn’t sound weak and an internal soundcard/External soundcard/External DAC (Digital to Analog Converter, a necessary step for all digital audio) would just improve the haziness/mud so you would hear details better, and depending on the brand the sound could be more lively and musical instead of “cold” and “matter of fact.” Treble is also often fatiguing with onboard audio – to be fair, this is also often an issue with cheaper soundcards/DACs too, but better gear will be less fatiguing and more pleasant, and once you hear it sounding great you won’t want to go back.
Another thing that is good to understand is that a soundcard serves three (or more) functions: Amp, DAC, and DSP (Digital Signal Processor). A DSP is kind of like a CPU, but dedicated to audio effects. Onboard audio may offer a limited EQ or reverb effect, but a DSP in a soundcard can actually distinguish between sounds that are supposed to come from in front of you, or behind and above, and also offer more intricate custom and pre-made EQ effects. I love surround sound when playing games or movies, and while the realism of surround improves with better headphones I still think it is a literal game-changer even with cheap headphones. A better DAC is responsible for better clarity vs haze, as well as lively vs dull. The Analog section feeds directly into the Amp, and a good amp can bring out the designer-intended performance out of a headphone, as well as affect the timbre of the sound.
Think of this: onboard audio is just a tiny chip or section of circuitry, a small portion of the overall budget that went into building your computer (or phone), while a soundcard is a dedicated piece of audio gear, with more room for the designs and upgraded components. Some people buy a Soundcard just to use as a DSP, and then connect a high-end external Amp and/or DAC to upgrade the sound even further, but most Soundcards are a good all-in-one place to start.
Alright, so hopefully that gives you an idea about the difference between integrated and discreet audio components. The story isn’t too different from graphics cards, huh? Now, since this rabbit hole has practically no bottom with “the best,” the more important question is “where are you happy?” If you are happy and content using your headphones with the onboard audio, then you don’t need to change. However, if you feel like something is missing or flawed, now you know that a soundcard is one step on the upgrade path. If you do want an upgrade, I would suggest that you will save money in the long run if you pick better than the cheapest so that you get something noticeably better than onboard audio, and focus on something with a digital output (like Optical) in case of a future where you want to try an external DAC some day but you still want to keep the benefits of the DSP you invested in with the soundcard.