Well, I’ll back up a bit and quickly sum up some other things. Souls have reiatsu, their projected spiritual energy. Particular characteristics or training can effect the quality of reiatsu, which sometimes is and sometimes isn’t related to the quantity a person has. We know how important strong will, determination, and focus are from canon. And of course, those that are trained and have good control can do more with less.

And it can have a direct effect on other souls. Heavy spiritual pressure can almost seem like gravity increasing or can overwhelm a soul with despair over the power difference. Or just straight up knock people unconscious.

Combined with someone who’s furious or uncaring of insignificant bugs, just being near them when they aren’t restraining their reiatsu can be horrifying.

But killing intent I think is something all its own. It doesn’t require immense spiritual power to back it up, though that helps, and it isn’t fueled by particular emotions, though they can be present.

It's essentially a projection of what one person intends to do to the other - kill them - the projected, prophesied result if the two (or more) individuals fought. It's a very specific dominance move, but not necessarily malicious. It /can/ be malicious, but it can also be cold, dismissive, uncaring. It can just be a warning of immovable reality - you will die if you pursue this course of action. It's the projecting of absolute confidence and surety of winning/domination, the projection of the death the opponent will face, in order to warn off, unbalance, or straight up terrify the target.

The way others experience it depends on the user’s control and choices - for example, projecting a very physical feeling, such as being choked to death, the throat being cut or torn out, stabbed in the chest or guts -- or it can be purely a mental attack; unfathomable doom, the instant awareness of the unbridgeable gap between user and victim, extreme weight/pressure, etc. Or suddenly understanding they have done wrong/sinned against a higher being, etc.

I like to imagine that the victims of Yamamoto using killing intent start to feel like they're burning alive. While Unohana 'puts the fear of god' so to speak, into the people that piss her off. Rukia’s probably feels cold. So it is part of reiatsu, in that that’s how it's formed and delivered to others - but it is also it's own thing, not just a projection of the emotions the soul happens to be feeling. It’s a weapon all its own. It's like.... a serial killer whispering in your ear how he plans to kill you in horrible and awfully specific ways, combined with the assurance that he is very much capable of doing it, and just how far he's willing to go to make it happen. This intent can be sharpened, dulled, lose or gain strength, the focus narrowed or broadened - it can be a blade under someones chin, or a miasma of death over a battlefield. Emotions like anger can either strengthen or weaken it depending on the user and the target.

Notably, it’s usually the most useful against less trained or less resolved opponents, and experience and setting one’s own resolve lessens the effect of killing intent. However it’s also standard in higher level fights, effects the quality of their reiatsu in attacks, even if it can’t convey other benefits.

But not all killing intent is equal as mentioned above - for example, facing off the killing intent of a mad scientist like Kurotsuchi Mayuri who wants to open you up and look at your insides is no doubt entirely different than someone who just wants to kill you quick and move on.

Add that to some mental manipulation? And you get the likes of Aizen and Gin, who can get into people’s heads and tear them down.

And of course, Kenpachi, who is the embodiment of killing intent, essentially uses it to filter out the weak and useless and scare out any hidden strength to make battles more interesting. Don’t you dare hold back in a fight with Kenpachi!