Hamilton was a hot-tempered bastard who received and handed out around a dozen duel challenges over the years. The vast majority of these didn't amount to much. "I'ma duel your ass" was mostly just frontin' -- an opportunity for both parties to give each other shit for a while, and let things cool down.
But in 1797, Hamilton found himself in a serious dueling situation. He was entangled in a government-fund misuse investigation and an adultery scandal, and became convinced that the future President James Monroe had something to do with his antics leaking to the press. So Hamilton sent Monroe a terse "heard you've been talking shit" letter, mentioning that Monroe should "bring a friend" when they'd meet -- a common euphemism for dueling. The angry Monroe promptly accepted, and when they met in person, the two men were so close to trading blows that they had to be physically restrained. The duel was on. To facilitate negotiations, Monroe appointed an old friend to act as an intermediary: Aaron Burr.
There were only around 20 people in those days, and they all knew each other.