Why was Tom's trial delayed?
Atticus explains that Tom's trial was delayed "to make sure there's nothing to be uneasy about." It is likely that tempers were high after Mayella accused Tom Robinson of rape and Atticus wanted the trial postponed so that people would not attack his client.
Atticus first mentions the postponement to Scout when he is explaining to her that he is defending Tom Robinson. He says, "It’s a peculiar case—it won’t come to trial until summer session. John Taylor was kind enough to give us a postponement." Scout says Judge Taylor "was a man learned in the law, and although he seemed to take his job casually, in reality he kept a firm grip on any proceedings that came before him."
Though Atticus doesn't say directly why he requested a postponement, he later says to Heck Tate that "we’ve gotten one postponement of this case just to make sure there’s nothing to be uneasy about." Since people are threatening Tom Robinson, it seems likely that Atticus was putting off the trial so that tempers could cool before Tom had to appear in court.
It's also possible that Atticus thought Tom would get a more fair trial by delaying until the summer session. Though it was unlikely from the beginning that Tom would be acquitted, Atticus had to do everything possible to help the innocent man go free. Ultimately, of course, he was found guilty and killed by prison guards.