The rules of upper class masculine society meant that insults, perceived or deliberately issued, had to be 'satisfied' by means of a duel. Thinking that Prince Andrei has insulted him, Rostov rashly attempts to provoke a duel to soothe his wounded pride.
Duels were traditionally provoked by the insulted party throwing a glove down in front of the insulter (hence 'throwing down the gauntlet'). They were a feature of Western society from the middle ages until the beginning of the twentieth century, and developed into highly ritualised occasions with strict rules. Perceived insults resulted in one man demanding 'satisfaction' from another, challenging them to an engagement in which they would have matching weapons, usually swords or, after the eighteenth century, pistols. The aim was to defend one's honour, so refusing a duel, or failing to demand one after an insult, caused deep disgrace.
Once a duel was inevitable, both combatants would appoint 'seconds' to arrange the time, place, choice of weapon and other details between them. Later the seconds were also charged with attempting a reconciliation between the parties. Duelling was banned in most countries, including Russia, although this did little to deter potential combatants. As time passed, the ritual aspect of duelling became more and more apparent, so that the aim often became to salvage the honour of the parties involved with the minimum of bloodshed. Having said that, people could still be killed in duels, as was Alexander Pushkin, Russia's foremost nineteenth century poet. Somewhat prophetically, in his most famous work, Eugene Onegin, the protagonist Onegin kills his best friend Lensky in a duel.