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