Extreme dexterity/control of the body, especially the hands and fingers that is adaptable to a wide range of movements.
You will never be able to teach a chimp for example, how to excell at ballet, action pistol shooting, gymnastics, etc…
Our hand-eye coordination is probably among the best too. Teaching a primate to hand write or draw portraits well would probably be very frustrating.
The sheer variety and complexity of sounds that humans are capable of making is probably unmatched. Some birds are excellent mimics, and many animals can make sounds that we can’t. But people can mimic an enormous range of sounds, not just with vocal chords but by whistling, clucking our tongues, snapping fingers or clapping our hands.
Our ability to hate & kill one another for any reason we like, or no reason at all.
What other animal does that?
Oh and porn too. We have been drawing dicks & boobs on everything since we learn we could.
…farting, burping…
This “What If” article claims that humans are uniquely good at accurate and precise throwing. It says:
Dolphins can toss volleyballs and frisbees with excellent aim.
ETA: Yes, frisbees. They fling them off the end of their snout, which doesn’t seem very aimable, but they can do it.
More ETA: And archerfish shoot bugs off of overhead branches by shooting water pellets at them.
Yeah? And assfish don’t need no stinking lungs at all.
Humans invented the wheel, New York, wars, and so on.
Language skill is a big biggie.
Roger Brown, a linguist of some note during the 1970-1980 era when there were a lot of animal language projects with chimpanzees and others, pointed out that advanced language skill makes knowledge cumulative over generations. No longer does each generation have to rediscover one lifetime’s worth of life skills afresh. But instead, each generation inherits the life skills of its predecessors through language, and each generation can build on that to develop new life skills to pass on to subsequent generations. Cultural development (which includes all facets of culture, science, technology, etc.) can take off and grow at an unparalleled pace not seen anywhere else in the animal kingdom.
I’ll turn around a previous poster’s comment, and say that we’re awfully good at socializing. We’re not unique among animals for that, no, but with that and our memory/intelligence and our endurance – messing with even one human is a good bid for a death sentence. Other humans will gang up on the perpetrator, who probably won’t survive.
Sure enough, we often enough turn on each other. We’re not unique among animals for that, either, however despicable our motives sometimes are.
However, dolphins aren’t remotely as good as humans. And archerfish can’t do it with anything but water.
And we’re extremely good at repressing our aggressiveness when necessary. I read a article recently - I think it might have been in Scientific Amerian that pointed out that 500 humans can board an airplane and sit in crowded and uncomfortable conditions for many hours without a single fight breaking out. With most other animals, half of them would be dead when the aircraft arrived if they were subjected to such conditions.
What else was there to do then? 
Not too many, but not none. Admittedly, no other animals use tools as complex as those that we make. But quite a few animals use primitive tools.
Primates use tools for hunting, gathering food, and even for fighting each other. Bonobos and chimps will sometimes make moss into a “sponge” of sorts that they use for giving themselves sponge baths. Gorillas have been seen using sticks to measure water depth.
Elephants create fly swatters and back scratchers.
Crows are smart little buggers. They create tools out of twigs and occasionally bits of wire, and have even been observed to make toys for themselves (which is very rare among animals).
More details here:
Chimpanzees.
Not only do small groups of chimpanzees fight and kill each other like human gangs fighting each other, but chimpanzees have even engaged in large scale warfare (this was quite a shock to Jane Goodall, who thought that only humans waged war).
Smell. No, not sense of smell, but odor. One theory about human evolution was that humans smelled so bad that predators avoided them.
There’s actually a number of examples of animals modifying found objects into tools. Wikipedia has a good summary of them. Admittedly, none have invented the iPhone yet, but give 'em time … give 'em time…
You can convince a human to pay a dentist to inflict incredible pain on them, in order to avoid more painful problems in the future. You can try to convince a chimp of this, one trained in sign language, but the chimp will always bite the dentist. So yes, I agree.
On a similar note, you could convince a human to undergo short term pain for long term gain, for instance, make an army rush to defend a certain point because otherwise you will lose. When this occurred hundreds of years ago, when many soldiers rode horses… well let’s just say you couldn’t explain to the horses to hurry.
Humans live on six continents, ranging from areas hot to cold, humid to dry and even changeable. This is mainly due to our intelligence and tool use, of course. Just picture an animal from the African savanna trying to survive in the arctic. It can’t. Humans were able to do so. They saw some animals with thick skins and fur, killed them, took those skins and fur and made warm clothing out of them.
Interesting.
We have the ability to track what other people are looking at, something only dogs (and not wolves) are good at. We’ve apparently evolved for it; both we and dogs are unusual in having lots of visible whites to our eyes, which makes tracking where someone is looking easier.
We are unusually good at locating the direction of a sound, and isolating one sound out of a cacophony. Which is why we can listen in to or participate in a conversation in a crowded room full of other people talking.
Humans are very good at heat endurance, probably a side effect of our adaptation for endurance running.
We’re good at finding and removing skin parasites, due to having hands and no fur.
As for violence, humans are actually much *less violent than most animals, including animals with mild reputations like deer. Humans aren’t especially violent; *what we are is lethal. Our skills and weapons outpace our instincts; we kill in incidents where other animals would only mildly injure one another.
Notice how unusual it is for one human to beat another to death with their bare hands, and how it’s (accurately) considered indicative of extreme brutality on the part of the beater? It’s hard to kill with your bare hands, especially accidentally; pick up even simple weapons like a rock, knife or club though and it’s much easier to kill with a momentary impulse.
It’s not just what others are looking at. No other animal besides humans and dogs understands a pointed finger intuitively, either. The other great apes can be taught to understand pointing, but only with great difficulty. This, despite most animals having clearly visible appendages which could easily be used for pointing.