Atticus, as a character, is a bit of a Southern anomaly for this time period -- that is, he is broad-minded, intellectual, objective, and incredibly just in all of his dealings, even with his children. In this way, he is well ahead of his time mentally speaking.
One struggle he faces is that of being a single father. Having to raise tomboyish Scout and her brother Jem in a time that was marked by the stereotypical nuclear family is a trial in and of itself. With no wife present, Atticus is left to play the role of both mother and father, with a great deal of help from Calpurnia, who serves as maid, cook, and mother-figure for the Finch children.
The second struggle that Atticus faces comes in the form of community perception. Prior to Atticus agreeing to defend Tom Robinson, he was thought of as a pillar of the Maycomb County community -- an unequivocable voice of fairness and reason as an attorney, and a member in good standing of the town itself. When Atticus agrees to defend Tom Robinson, his reputation and his community standing are both brought into question, and furthermore, are greatly disparaged. This disparagement by his neighbors, friends, and society at large comprises Atticus's second struggle as a character.