Burr was a dangerous conservative who wanted nothing less than to be crowned king. He was an enemy to Democracy, and Hamilton knew it.
Hamilton asked for a duel over a minor slight b/c he knew that Burr would not fight like a gentleman. (Traditionally a gentleman who was challenged would fire the first shot into the air, then both sides would reconcile.)
Both sides fired. Hamilton shot into the air; Burr, being the typical conservative, shot to kill. However, Hamilton also knew that if Burr murdered him that it would be the end of Burr's political career.
Burr was a coward, war profiteer and an enemy to the republic. In other-words, he would have made an upstanding modern-day Republican.
For more info on Burr:
http://www.ushistory.org/valleyforge/served/burr.html