The Mensheviks and Bolsheviks split during the Russian Social-Democratic Labor Party's Second Congress in 1903. The initial split was over some relatively minor procedural disagreements in the functioning of the RSDLP, but the leadership between the two factions, as you'll see, ended up having some fundamental disagreements.
The Bolsheviks, led by Lenin from the beginning, believed in a much more proactive revolutionary agenda and had more support among the working class in the Russian parts of the Russian Empire (read: Petrograd and Moscow). By proactive, I mean the Bolsheviks were far more steadfast in their belief that a socialist revolution had to be spurred by a class of revolutionaries (a revolutionary vanguard) that would be the tip of the revolutionary spear, as it were.
To wit - Lenin wrote in 1903, in his summary of the Second Congress " The Russian Social-Democratic movement is in the throes of the last difficult transition from the circles to a Party, from philistinism to a realisation of revolutionary duty, from acting by means of scandal-mongering and circle pressure to discipline." (https://www.marxists.org/archive/lenin/works/1903/sep/15a.htm)
After 1903, the Mensheviks were led by Trotsky, Martov and Plekhanov, and their faction took a much more orthodox Marxist approach to revolution. Marx's understanding of history was that the overthrow of capitalism was part of a historical trend that would happen as a result of capitalism's inherent contradictions. As such, they were not as interested in driving a revolution through the work of a revolutionary vanguard. The Mensheviks also drew more of their support from the non-Russian parts of the Russian Empire.
The bases of support were quite important, especially by the time of the First World War, when those distinctions were amplified by a working class that was increasingly frustrated by the Tsar (and his war). As the more explicitly militant faction, the Bolsheviks had an easier time mobilizing the disaffected working class of Russia's capitals.
Remember, of course, that "the" Russian Revolution in 1917 was really two revolutions - February and October. The February Revolution ousted the Tsar and installed a provisional government that was to lead to elections, but it was not until October than the Bolsheviks seized power. The intervening months provided a series of events that I won't go into at length here (July Days, Kornilov Affair), that greatly favored the Bolsheviks more militant approach.
It can't really be stated enough that the defining feature of the Lenin-led Bolsheviks prior October 1917 was that it didn't just predict a proletarian revolution, it was actively trying to make it happen. That didn't mean that a figure like Trotsky, who was a Menshevik leader, couldn't be brought into the revolutionary fold in 1917, but Trotsky was not spending his time between 1903 and 1917 building a working-class revolutionary vanguard the way Lenin was.
The great news is a lot of the primary source material here available in translation for free at http://marxists.org - for example, you can see Lenin's writings from 1903, the year of the RSDLP's Second Congress, here: https://www.marxists.org/archive/lenin/works/date/1903.htm
I think it's really important to understand the Bolshevik / Menshevik split in the context of revolutionary politics in Russia at the turn of the century, so if you're interested in further reading, there are a number of excellent works on the revolution in general that you can consider depending on how broad you want to go:
Sheila Fitzpatrick's The Russian Revolution.
Ronald Grigor Suny's The Soviet Experiment
Mark Edele's (brand new) - The Soviet Union: A Short History
Robert Service's - A History of Modern Russia