The Red Terror was a program of mass repression, class extermination and execution carried out by the Bolshevik government during the Russian Civil War.
The Russian Revolutions
In 1917 several decades of institutional decay, chronic mismanagement, rising political awareness, and a terrible war caused the Tsarist regime in Russia to be confronted by such a large rebellion, including the loss of the military's loyalty, that two parallel regimes were able to take power in Russia: a liberal Provisional Government, and a socialist soviet. As 1917 progressed the PG lost credibility, the soviet joined it but lost credibility, and extreme socialists under Lenin were able to ride a new revolution in October and take power. Their plans caused the start of a civil war, between the Bolshevik reds and their allies, and their enemies the Whites, a large range of people and interests who were never properly allied and who would be defeated because of their divisions. They included right-wingers, liberals, monarchists and more.
The Red Terror and Lenin
During the civil war, Lenin's central government enacted what they called the Red Terror. The aims of the were twofold: because Lenin's dictatorship seemed in danger of failing, the Terror allowed them to control the state and reforge it through terror. They also aimed to remove whole classes of state 'enemies', to wage a war by the workers against bourgeois Russia. To this end, a massive police state was created, which operated outside the law and which could arrest seemingly anyone, at any time, who was judged a class enemy. Looking suspicious, being in the wrong time at the wrong place, and being denounced by jealous rivals could all lead to imprisonment. Hundreds of thousands were locked up, tortured and executed. Perhaps 500,000 died. Lenin kept himself apart from the daily activity like signing death warrants, but he was the driving force that pushed everything up the gears. He was also the man who canceled a Bolshevik vote banning the death penalty.
Channeling the Anger of the Russian Peasants
The Terror wasn't purely a creation of Lenin's, as it grew out of the hate-filled attacks which vast quantities of the Russian peasants directed against the perceived better off in 1917 and 18. However, Lenin and the Bolsheviks were happy to channel it. It was given a great deal of state support in 1918 after Lenin was nearly assassinated, but Lenin didn't redouble it simply out of fear from his life, but because it had been in the fabric of the Bolshevik regime (and their motivations) since before the revolution. Lenin's guilt is clear if once denied. The intrinsic nature of repression in his extreme version of socialism clear.
The French Revolution as Inspiration
If you've read about the French Revolution, the idea of an extreme group introducing a government that ran through terror might seem familiar. The people caught up in Russia in 1917 actively looked to the French Revolution for inspiration - the Bolsheviks thought of themselves as Jacobins - and the Red Terror is a direct relation to The Terror of Robespierre et al.