The answer of Sola Gratia is obviously correct but does not explain why Haman's hatred was spurred by the fact that Mordechai was Jewish. If Haman only wanted to remove an insufficiently obsequious subordinate he could have easily had him removed; however, he uses this occasion for genocide. Why such an extreme reaction?
Note that Haman was called an "Agagite", that is, a descendant of "Agag". This is almost certainly a reference to the incident in 1 Sam 15 where King Saul was supposed to exterminate the Amalekites but spared the best of the livestock and the King, Agag. When the prophet Samuel went to Saul to rebuke him they clashed bitterly, and parted never to see each other again (1 Sam 15:35).
However, Samuel had King Agag brought before him and executed him by cutting him to pieces in public view (1 Sam 15:32, 33). One presumes that Agag had a son who survived this whole affair as Haman traced his ancestory back to King Agag and obviously nurtured a long-held grudge and used Mordechai as an opportunity to reap revenge on the Jews by exterminating them.
Note the story also told by the historian, Josephus (Antiquities of the Jews, Book XI, chapter 6 section 5) also explains Haman's grudge against the Jews because Haman was an Amalekite which nation had been exterminated by the Jews.