Language is what makes us human. The use of language is a foundational aspect to the formation of culture. Culture is what separates us from the animals. We are human because we can communicate and remember. Language gives us both those abilities.
I agree with lrwilliams. Humans are the only species capable of communication through verbal use of words. It is also said, that humans are also the only species that use sex as an emotionally or purely physically pleasing activity rather than merely the means of procreation. That being said, I was not aware that other species have no cerebral cortex, and I wonder if this is true. However, other species have been shown to have an ability to use higher order thinking to solve problems.
The human brain is significantly larger than that of all other animals, including the animals that are similar to us in size. A recent study by Harvard scientists did show that some animals can think in ways that were once considered to be unique to humans, such as having an episodic memory.
However, the study would further show that there are still cognitive rifts between humans and other highly intelligent animals. For instance, other animals may be able to use a specific solution to solve a specific problem; however, only humans are able to apply solutions to new situations or to solve different kinds of problems.
Another example is that other animals, such as chimpanzees, have the ability to use simple tools. But only humans can combine materials to create a tool with different functions.
One thing that comes to mind that makes us unique as humans is our ability to communicate through the use of speech. While other animals communicate through other forms, I believe humans are the only ones which use language.
As Post #2 does, I think you need to clarify what you mean by this question. Are you asking what makes human beings unique among the animals or what makes each person unique among humans?
I would take issue with Post #2 about what makes us each unique. If he's right, then identical twins aren't really unique. I think it's not DNA that makes us unique. I think that it is our experiences (combined with our DNA) that make us unique. No one has exactly the same experiences as anyone else and therefore each of us is unique.
There's two ways to look at this question. First, what about me as a human differentiates me from different species? In that case, humans are the only species on Earth that have the ability to reason, due to our cerebral cortex, which is lacking in every other species. While other species do have intelligence, they do not have the ability to reason and solve complex problems.
Second, what differentiates me as a human from other human beings? In that case, I am the result of a complex genetic code, a DNA that is truly unique. There is no one else on Earth that is exactly like me, either in appearance or personality. The sheer number of possible combinations of genes makes all 6.5 billion human beings unique from all others.