Bolsheviks
Lenin's faction of Russian Marxists, who took their name from a single lost vote in 1903 and the Russian state fourteen years later.
The name is a piece of branding that outlived the argument that produced it. At the Second Congress of the Russian Social Democratic Labour Party in 1903, held partly in Brussels and partly in London because the tsarist police made meeting in Russia impossible, Lenin's supporters won a narrow vote on the composition of the party newspaper's editorial board. Lenin immediately began calling his group the bolsheviki, the majority men, and his opponents the mensheviki, the minority men. In the years that followed his faction was frequently the smaller one. The label stuck regardless, which tells you most of what you need to know about how the Bolsheviks operated.
What the split was actually about
The dispute looked procedural and was not. It concerned who counts as a party member. Julius Martov wanted the party open to anyone who supported it and worked under its direction, on the model of the mass socialist parties of western Europe. Lenin wanted membership restricted to those who personally participated in one of its organizations, which in a police state meant a small body of full-time professional revolutionaries.
Lenin had laid out the reasoning in What Is To Be Done? in 1902. Workers left to themselves, he argued, develop trade union consciousness, meaning they fight for better wages and stop there. Revolutionary consciousness has to be brought to them from outside by a disciplined vanguard. The implication is the important part: if the party embodies the true interests of the working class, then the party may overrule the actual working class when the two disagree.
1917
The Bolsheviks were a marginal group in early 1917, and most of their leadership was in exile or Siberia when the tsar fell in February. Lenin returned in April, crossed Germany in a sealed train arranged by a German government that correctly expected him to be disruptive, and stunned his own party by refusing any cooperation with the Provisional Government. The slogans he settled on, peace, land, and bread, and all power to the soviets, addressed the three things that actually moved Russians in 1917, and no other party offered all three.
On 25 October by the old calendar, 7 November by ours, Red Guards and sympathetic soldiers took the bridges, telegraph offices, and rail stations of Petrograd, and then the Winter Palace. It was less a storming than an occupation of a building nobody was defending. Holding power took three more years of civil war, roughly seven million dead including famine and epidemic, and the creation of the Cheka, the first Soviet secret police, in December 1917.
What happened to the name
In March 1918 the party renamed itself the Russian Communist Party (Bolsheviks), and the parenthesis stayed in the official name until 1952. By then it described nothing. The faction that had defined itself by internal argument had banned factions at its own Tenth Congress in 1921, and every Bolshevik of Lenin's inner circle except Stalin was dead by 1940, most of them shot by the party they founded.
Why the model traveled
The vanguard party was Lenin's real export. Through the Comintern it was copied in China, Vietnam, Cuba, and a dozen other places, usually with the tactical lesson attached: a small, disciplined, ruthless organization can take a state during a collapse even when it does not have majority support, because in a collapse nobody has majority support. The organizational chart of the Chinese Communist Party today is a direct descendant.