One of the first impressions of Tom's character that comes out in chapter two is that he was not very considerate of the feelings or opinions of others. Tom's actions demonstrated his pride in having a girlfriend more than they displayed true devotion to her.
His acquaintances resented the fact that he turned up in popular restaurants with her and, leaving her at a table, sauntered about, chatting with whomsoever he knew.
In the same way, Tom didn't give Nick any choice about the timing, circumstances, or location of his introduction to Myrtle.
his determination to have my company bordered on violence. The supercilious assumption was that on Sunday afternoon I had nothing better to do.
Late in the evening, Tom's true nature as a bully, aided by the generous amount of whiskey consumed during the time they were in the New York apartment, was displayed. He and Myrtle got into an argument regarding whether or not she should mention Tom's wife, Daisy, when they were together. Tom obviously contended that pointing out that he was married to someone else was not necessary; Myrtle apparently didn't see any reason why she couldn't discuss Daisy whenever she felt like doing so.
"Daisy! Daisy! Daisy!...I'll say it whenever I want to! Daisy! Dai-" Making a short deft movement, Tom Buchanan broke her nose with his open hand.
There are a number of quotes in Chapter Two which reveal important details about the character of Tom Buchanan. Early in the chapter, for example, is the following quote about Tom's relationship with his mistress:
"The fact that he had one [a mistress] was insisted upon wherever he was known."
This quote clearly demonstrates Tom's arrogance. It shows that he believes it is acceptable to flaunt his mistress around town, even though he is a married man and word will certainly get back to his wife, Daisy. Tom clearly has a strong sense of self-entitlement which enables him to live his life however he chooses, regardless of the cost to others.
Secondly, this quote is also important in understanding Tom's character:
"Making a short deft movement, Tom Buchanan broke her nose with his open hand."
As this shows, Tom's physical strength is easily transformed into violence and he is unable to control his temper. That Tom hits Myrtle for speaking Daisy's name shows that his personal loyalties are often contradictory. He is prepared to disrespect Daisy by having an affair, for instance, but will not allow his mistress to treat Daisy in a similar fashion.