Who understood Gatsby the most?

By the end of the novel, the one who best understands Gatsby is Nick.

Daisy loves Gatsby, but she does not know much about who he is or where he comes from. Nor does she realize, when they are reunited, that he thinks she will now divorce Tom and pick up her romance with Gatsby where it left off. When Tom brutally reveals to her that Gatsby is not who he claims to be, she is shaken and confused.

Tom "understands" Gatsby in the sense that he realizes Gatsby is not what he seems. He doesn't trust Gatsby's word, and does some research on where his money is coming from. Tom understands Gatsby's economic and social status better than anybody, but he does not understand Gatsby's thinking or motivations.  

Nick does not start out understanding Gatsby well. At the beginning of the novel, he has never met him. Then, attending a party of Gatsby's, he hears rumors about him, but still doesn't know what he looks like or how old he is. He talks with Gatsby for several minutes before he realizes that the young man he is speaking with is Gatsby.

As the friendship develops between the two men, Nick often gets the feeling that Gatsby is not telling the whole truth about himself and his past, but then Gatsby will offer some kind of proof, and Nick will immediately trust him again. But he still knows very little of Gatsby or his motivations.

It is not until Gatsby asks Nick, through Jordan, to arrange a meeting with Daisy, that Nick finds out Gatsby's reason for the huge house, the location of the house, and the wild parties. Jordan tells Nick a story from five years ago that reveals that Gatsby and Daisy were once in love. At this point, Nick feels that now Gatsby makes sense to him. He still does not know how much of what Gatsby has told him is true, but he understands Gatsby's feelings well enough to help him get over his nerves and embarrassment when he first meets Daisy. Nick (and, it seems, Nick alone) realizes how much of an emotional investment Gatsby has made in the idea of Daisy, and the idea of himself living in Daisy's world as Daisy's lover. 

During the awful argument between Tom, Daisy, and Gatsby, Nick and Daisy realize for the first time that Gatsby really is mixed up in illegal business. It's after this that Daisy and Gatsby, racing home from that awful scene, hit Myrtle Wilson with their car. Nick comes in the other car with Tom and witnesses the havoc of Myrtle's death. When he next sees Gatsby, he is disgusted with him, in part because he thinks it was Gatsby driving. Then he finds out that it was Daisy driving—and it's on that same evening that Gatsby finally tells Nick his personal history: how he was born James Gatz of North Dakota, how he seduced Daisy without ever intending to fall in love with her, and so on. At the end of this conversation, the exhausted Nick still does not like Gatsby very much, but he understands him very, very well.

After Gatsby is shot, Nick's sympathies shift. He feels he is the only friend left to the dead Gatsby. He becomes even more loyal to Gatsby when he sees that none of Gatsby's former "friends," including Tom, Daisy, and all the people who attended Gatsby's parties, will so much as come to his funeral.

The only exception is one man whom Nick once met at a party of Gatsby's. This man, at the time, was in Gatsby's library, drunk and marveling that all the books were real. Though not as good a friend as Nick, this man arguably understood Gatsby a bit. He sensed Gatsby was a fake, hence he expected to find the books were fake. When he found they were real, he was impressed, perhaps sensing that Gatsby was trying not just to fake at being an upper-class person, but to actually turn himself into a different person.