You are going to hear this week that Mordecai refused to bow to Haman because Haman wore an idol around his neck and Mordecai wanted it perfectly clear that idol worship was wrong.
That's one Midrash, and a popular midrash, but its not the only view on the subject. The Yalkut Shimoni 956, for example, says that Mordecai refused to bow because he was a prisoner of his own pride. Haman, the midrash says, was actually Mordecai's slave, and Mordecai could not bring himself to show him any reverence. In the view of this Midrash, Mordecai's vanity created the crisis and his teshuva ameliorated it. (This parallels Moshe who first abandoned klal yisroel before returning to redeem them) On the other hand, Esther Rabbah 7:8 says that the Chachamin of the time wanted to bow to Haman, and follow the precedent of Yaakov who bowed to Esav. They thought this was how a galus Jew was supposed to behave. Mordecai, as discussed in the post directly beneath this one, held differently.