The eyes of Dr. T. J. Eckleburg can be seen as a symbol of an all-seeing God. This remarkable piece of advertising, displayed on a decaying billboard in the Valley of Ashes, stands as a constant reminder that, no matter what we do, God sees everything. The Almighty may not play a large part in The Great Gatsby, but he's there all the same, watching over the various characters as they engage in all manner of appalling behavior.

None of the characters in the story appear to pay more than lip service to the belief that God exists. For Gatsby, wealth and social acceptability are his personal deities, at whose altars he regularly worships.

As for the Buchanans, high social status is their god, which explains why Daisy, despite conducting an affair with Gatsby and telling him that she loves him, is not prepared to ditch Tom for Jay.

When reading The Great Gatsby, it's important to remind ourselves that it was only a tiny, privileged elite that led the kind of hedonistic, materialist lifestyles led by the crowds in West and East Egg. Many Americans remained committed, with varying degrees of consistency, to traditional God-fearing values. Many would've looked askance at the reckless, immoral behavior of the glittering social milieu presented to us in the story. To some extent, then, the eyes of Dr. Eckleburg don't just represent God, but the traditional values that many Americans followed in the Roaring Twenties.

F. Scott Fitzgerald's novel The Great Gatsby is full of symbolism, and the eyes of Dr. T. J. Eckelburg are an important example of this. In the Valley of Ashes, a desolate and ruinous place on the way from West Egg to Manhattan, there is a billboard advertising the services of an oculist (which is what optometrists were called in the 1920s). Nick Carroway describes the billboard as a huge pair of blue eyes wearing yellow glasses and without any other facial features. The condition of the billboard, weathered with age and dirty, suggests that Dr. T. J. Eckelburg is no longer practicing in the Valley of Ashes, but his omniscient eyes watch over the events of the people there. This billboard first appears in chapter two when Tom takes Nick to meet his mistress, Myrtle Wilson, a resident of the Valley of Ashes. 

But above the grey land and the spasms of bleak dust which drift endlessly over it, you perceive, after a moment, the eyes of Doctor T. J. Eckleburg. The eyes of Doctor T. J. Eckleburg are blue and gigantic - their retinas are one yard high. They look out of no face but, instead, from a pair of enormous yellow spectacles which pass over a nonexistent nose. Evidently some wild wag of an oculist set them there to fatten his practice in the borough of Queens, and then sank down himself into eternal blindness or forgot them and moved away. But his eyes, dimmed a little by many paintless days under sun and rain, brood on over the solemn dumping ground… I followed [Tom] over a low white-washed railroad fence and we walked back a hundred yards along the road under Doctor Eckleburg's persistent stare . . . "Terrible place, isn't it," said Tom, exchanging a frown with Doctor Eckleburg. 

Fitzgerald uses personification here to give human qualities to the billboard when he suggests that it shares a frown with Tom. 

The next time readers encounter the eyes is in chapter seven when they foreshadow the tragic events to come. 

Over the ashheaps the giant eyes of Doctor T. J. Eckleburg kept their vigil but I perceived, a moment, that other eyes were regarding us with peculiar intensity from less than twenty feet away. In one of the windows over the garage the curtains had been moved aside a little and Myrtle Wilson was peering down at the car.

In this chapter, Myrtle's husband has found out about her affair and is keeping her locked away. Myrtle is staring at the yellow car, thinking it contains Tom who will come to her rescue. Of course, this is how she meets her tragic end, by running out in front of the car. 

In chapter eight, readers learn that George interprets the eyes as the eyes of God, watching over his wife's actions in judgment. He proclaims that God—the eyes—watch her, and while she may have been able to fool him, she can't fool God. He says to Michaelis, the coffee shop owner, 

"—and I said 'God knows what you've been doing, everything you've been doing. You may fool me but you can't fool God!' "

Standing behind him Michaelis saw with a shock that he was looking at the eyes of Doctor T. J. Eckleburg which had just emerged pale and enormous from the dissolving night.

"God sees everything," repeated Wilson.

"That's an advertisement," Michaelis assured him. 

The eyes of Dr. Eckleburg are literally eyes on a billboard and figuratively witness to the comings and goings and the deeds of those who pass through the Valley of Ashes. 

...the symbolic eyes of Doctor T. J. Eckleburg [watch] over this “solemn dumping ground” as a God-like witness to the...hopelessness that emanates from the place.

While Myrtle and George Wilson, people of modest means, live within sight of this billboard in the modern pseudo-waste land between the suburbs and the city, characters like Gatsby, Tom, Daisy and Nick only pass through this depressed and depressing area. 

This region is where Myrtle is killed and where Gatsby's fate is sealed. 

The characters in the novel behave badly, acting without any clear moral sense. Though there is little mention of religion in the text, the eyes of Dr. Eckleburg are directly associated with the eyes of God. They stand as the symbolic witness and judge of the events that have taken place, leading to the deaths of two people. 

George Wilson draws this comparison after his wife is killed, confusing the eyes on the billboard with the judging eyes of God. 

Thus the eyes are a literal representation of commercialism (placed on an advertisement) and a figurative/symbolic representation of a silent force of judgement that finds its place at the end of the novel. 

The first mention of the the eyes of Dr. T. J. Eckleburg is at the start of Chapter 2.  When Tom brings Nick into the city and he meets Myrtle, Tom's mistress, Nick sees a part of New York that is completely unlike either East or West Egg, or the city of New York proper.  Nick calls this middle place the "valley of ashes" because everything in this area is run-down place is covered in ashes -- it is an industrial waste land where everything is very bleak.  In contrast to those images, Nick notices the eyes of Doctor T. J. Eckleburg.  He provides the following description:

[The eyes] are blue and gigantic -- their retinas are one yard high.  They look out of no face but, instead, from a pair of enormous yellow spectacles which pass over a nonexistent nose.  Evidently some wild wag of an oculist set them there .. [and now they] brood on over the solemn dumping ground. 

This enormous billboard, advertising for an eye doctor, looks out over the wasteland that is the valley of ashes.  The billboard is located across the street from Wilson's garage, where Myrtle lives with her husband.  The billboard is an interesting symbol because eyes usually represent sight, but these eyes are "blind," yet present -- perhaps a symbol that someone/God is watching everything.  God sees Tom and Myrtle carrying on their affair.  God knows what Daisy will do to Myrtle on this same piece of street when she races back from the city in chapter 7.  The color of the billboard is also symbolic.  Blue frequently represents dreams, but yellow frequently represents disease and decay, so the combination of the two implies that there are corrupt dreams here.  That can be seen by all of the characters:  Wilson wants more business and to achieve more wealth; Tom wants Myrtle; Myrtle wants a life-style that Wilson can't provide, but that she can pretend to have with Tom as his mistress.  Nick's final comment in this chapter is that they walked under Doctor Eckleberg's persistent stare.  To personify the billboard as if it would actually see clues the reader into the fact that those eyes are peculiar and potentially important.