I personally use Sennheiser HD518s for gaming and music.

You have on-ear vs over-ear. I prefer the latter for comfort, but if you have glasses it may be irritating since the ear pads could cause them to dig in and leave imprints. On the other hand it hurts my ears to have on-ear headphones on for too long.

Then you have closed-back vs open-back. The former will block out the sounds around you, the latter being open will allow sound to travel both ways. Open backed headphones are better for shooters because the open design allows for a wider soundstage, helping you locate where sounds are coming from more accurately. I do notice the difference personally when playing overwatch, closed headphones of similar quality (hd428) are more difficult to track enemy movement with.

I do not recommend you get gaming headsets, tried some and they are OK but not worth the price tag, you can do much better getting a quality set of headphones and pair it with a mic. If you have a webcam you can use that for a mic. Or get a cheap clip on mic for your headphones. Zalman is often recommended and is about $6 on Amazon.