There are three forms of irony, broadly speaking: dramatic, verbal and situational. In dramatic irony, the audience knows something important the characters don't. In verbal irony a character says the opposite of what he or she means: often this is the same as sarcasm, such as calling someone "graceful" when they fall down. In situational irony, slippage exists between what might be expected to happen and what actually occurs.
Nick is a narrator with a keen sense of irony. Most of the irony he describes in the first two chapters is situational. For example, he understands that Gatsby, a person he should despise, is the one he admires most, saying there was something "gorgeous" about him. Further, Nick notes that his college peer and now neighbor, Tom Buchanan, is extremely wealthy, marveling to himself that someone almost his own age could afford to own a string of polo ponies. Yet, in a democracy, in America, supposedly the land of opportunity and the self-made man, Nick doesn't miss the irony that someone like Tom, a person of limited intelligence who is also a racist, should have inherited vast wealth. In yet another irony, although one would expect Daisy, living in vast wealth in a red brick colonial mansion with every luxury at her disposal (the outward form of the American dream) to be happy, she is in fact discontented, in part because Tom is having an affair.
An example of verbal irony comes from Daisy in the first chapter. As Tom expounds on his embarrassing and outdated theory, imbibed from a book he is reading, that the "Nordic" race is superior but will be "submerged" by the darker skinned races, Daisy says, "Tom's getting very profound ... He reads deep books with long words in them." Lest the reader think to take Daisy at face value, Nick points to the irony in her words by saying "she winked at me again," as if the two share the secret that Tom is ridiculous.
In the second chapter, Nick again shows his awareness of situational irony, this time at Myrtle's apartment. She may put on airs and play the part of the grande dame, but in fact, Myrtle is a lower class woman trading sex with Tom for a few luxuries and actually with very little power, as is graphically illustrated when Tom hits her in the face and she can do little but tolerate it.
There are several, but here are a few of the more obvious ones:
Nick says that Gatsby embodies everything he dislikes, but then goes on to say how much he admires Gatsby.
Daisy and Tom seem to "have it all" yet Nick can sense Daisy is unhappy.
Daisy hopes that her female child would be born a fool.
Daisy knows that her husband is having an affair, but no one will say anything about it.
Gatsby is the central man in this novel (it's named after him), and his is barely in the first two chapters.
Myrtle's impulsiveness seems to entice Tom, yet, he punches her in the nose when she impulsively chants Daisy's name.
Nick seems more alert and interested at the subdued dinner party than at the wild party with Myrtle at the apartment.