The other answers for this question do a great job describing Nick. I agree that Nick is an observer, though not an impartial or dispassionate one. Part of what makes Nick a great character is that, while he observes what is going on around him, he is not always so good at observing what is going on inside him. It is true that he thinks of himself as "honest" and "non-judgemental," and I suppose compared to the likes of Tom he is. However, it's not clear from the novel whether he fully understands, say, his attraction to Jordan (beyond musing about the perspiration on her lip when she plays tennis) or his common feeling with Gatsby. By this I mean that what sets Nick apart as a narrator is his empathy for the other characters. He is watching, but he is also feeling. While he can observe with clear eyes what happens to Gatsby and Tom and Daisy, it is also clear that he is not separate from their fates, nor is he exempt from whatever forces may be at work to bring people together or tear them apart.  

There are two significant statements that Nick makes in the novel that, according to him, define his character and make him different. The first is in chapter one where he declares:

"I’m inclined to reserve all judgments..."

It is therefore painfully ironic that we find, however, that throughout the novel Nick does make unreserved judgments. When he, for example, describes Tom Buchanan, he uses phrases such as, "Tom would drift on forever seeking", "Two shining arrogant eyes", "gave him the appearance of always leaning aggressively forward", "a cruel body", " ... the impression of fractiousness he conveyed. There was a touch of paternal contempt in it." 

In a similar vein, he describes Daisy Buchanan, Jordan Baker, the guests at Jay Gatsby's parties, Jay Gatsby himself, etc. It is in chapter eight, however, that Nick's so-called "reserved judgment" is not so reserved after all, when he shouts to Jay:

“They’re a rotten crowd,” “You’re worth the whole damn bunch put together.”

"I am one of the few honest people that I have ever known." 

The remark comes across not only as high-handed and facetious, but also judgmental and self virtuous, for throughout the novel, Nick does a number of things which cannot possibly be deemed "honest." Firstly, he starts a relationship with Jordan Baker, knowing full well that he is still involved in a relationship back home, an affair he describes as, "that tangle back home." He conveniently dismisses this relationship by saying:

"Nevertheless there was a vague understanding that had to be tactfully broken off before I was free."

But he does not break off the relationship, opting to go along with what he has with Miss Baker. 

Furthermore, Nick prefers to remain silent about Tom's affair with Myrtle Wilson. He does not confront Tom about it and neither does he inform Daisy. He also does not say anything about Daisy being responsible for Myrtle Wilson's death. Once again, he prefers keeping mum.

This is the true irony. 

On the first page of the novel, Nick says he is very tolerant and tries to reserve his judgment, meaning he does not judge other people on terms other than their own. This quality makes Nick feel different, and at times superior to those around him who are obviously obsessed with wealth, materialism, and appearances.

The interesting point on which to focus, however, might be that although Nick proclaims that he is non-judgmental, he spends much of the novel judging the actions of others from his own moral standpoint.

In The Great Gatsby by F. Scott Fitzgerald, what quality does Nick possess that he thinks makes him different?

We learn the most about Nick Carraway, the narrator of The Great Gatsby by F. Scott Fitzgerald, in the opening pages of the novel. He tells us where he came from (the Midwest) and what he has been doing since college. He also tells us where he is living and why he is here. He also tells us about the advice his father once gave him. Nick says this about his father:

"Whenever you feel like criticizing any one," he told me, "just remember that all the people in this world haven't had the advantages that you've had."

Because of that, Nick is certain that he has a gift few other people have. He says,

[i]n consequence, I'm inclined to reserve all judgments, a habit that has opened up many curious natures to me and also made me the victim of not a few veteran bores.... Reserving judgments is a matter of infinite hope. I am still a little afraid of missing something if I forget that, as my father snobbishly suggested, and I snobbishly repeat, a sense of the fundamental decencies is parcelled out unequally at birth.

This concept of withholding judgments is significant for a storyteller because, if indeed it is true, the listeners (or readers) can trust whatever the narrator says. If he is not a reliable narrator, we (readers) are going to have to work hard to determine the truth of everything Nick says. Unfortunately, since Nick is the person telling us about himself, we have to wait until something actually happens in the novel to decide for ourselves whether Nick truly does "reserve all judgments," as he claims he does.

I would like to give you some examples from the story to examine, but by your question I presume that you are just beginning the novel and I do not want to ruin the story for you. Instead, here are some general guidelines for testing a narrator's reliability. First, examine what other characters say and do and then "listen" to how the narrator describes those same things. Second, determine what other people think and feel about the characters the narrator talks about to see if there is consistency. Finally, listen carefully to what else the narrator says about himself and see if he has a realistic and accurate view of himself based on what you see and hear. 

What quality does Nick possess that he thinks makes him different in The Great Gatsby?

Nick was told by his father, at some point before the time of the adventures retold in The Great Gatsby, that "a sense of the fundamental decencies is parcelled out unequally at birth" and that he, Nick, should remember that he is advantaged over many others in this regard.

Nick accepts this statement from his father as being true in his case, and sees it as being an important influence in shaping the way in which he interacts with other people. In particular, Nick considers himself more open-minded and less judgmental than most other people. This is a mixed blessing at times, as he comments,

I'm inclined to reserve all judgments, a habit that has opened up many curious natures to me and also made me the victim of not a few veteran bores.

Overall, however, Nick considers his openness to whatever personality quirks he observes in others as being a unique asset in his personality.