People smarter than me call it "kinaesthetics". Some call it "game feel". Basically, the game needs to use sound, animation, response to damage, and input delay together in order to help transcend the reality that I'm just a guy holding a controller.
For a bad example, any Assassin's Creed game. The animations are so long and it feels so floaty, that I'm not even sure I have 1:1 control of the character for fucking anything. It doesn't help either that even the joy of platforming is reduced to holding down the "awesome trigger" and moving the stick in a direction. This is why I've never played more than an hour of any of them.
It's a really specific and not fully understood thing, which is why those who nail it get accolades for it. In Demon's Souls, the way it felt to have L1 lift up your shield, and the ability to do a rhythmic ebb and flow in order to manage your stamina while also blocking attacks was fucking amazing, and in the grand scheme it's a small thing. My cousin (who had never seen a Souls game before) was watching me go through No-Man's Wharf with the archers, and I pulled up the shield right as an arrow was about to hit me, and he was like "whoa, that was cool that you saw it coming at the last second and could just whip the shield up like you would for real".
This is going to sound strange, but noises/sound effects are major to improving combat, even at the expense of "realism". In Quake 3 Arena, when your shots hit there was a satisfying "pwomp" when your shot landed, confirming for you that it did in fact hit to help you plan out the rest of the attack/retreat. In more modern games, the enemy needs to groan, or something audible and also react with animation to make it feel right. For all its failings, Rage fucking nailed it when it came to enemies taking your shots. Condemned: Criminal Origins is still to me the pinnacle of weighty and impacting first-person melee.
It's become the bane of modern open-world games right now (I almost made a thread about this), but developers have taken notes from the Arkham games in this regard. The thing is, it's not enough to copy the rhythm game style of racking up a combo and ice skating from enemy to enemy with rhytmic taps and occasional counters; what made Arkham work was that the hits had impact, the elbows and punches and the way that the enemies looked like they got wrecked by it. It was a smart move, having the player perform rhythmic taps that caused batman to gracefully slide from enemy to enemy with that rhythm, but with bone-crushing impact to make the hits satisfying. I do hope we see an evolution or replacement of this system though, because it's become the go-to melee system for open-world games and it doesn't fit all scenarios. Adding on more gadgets is only extra whipped cream on the sundae.
Oh, and this might go without saying, but QTEs are the opposite of satisfying combat. I'm playing a shit game of Simon Says in order to trigger an animation, or worse, prevent me from "failing a cutscene" that must be restarted. I would have enjoyed RE6 a lot more (I liked the melee/stamina management, the combat roll, etc.) if there weren't SO MANY instances of this kind of QTE littered all over every campaign.