Trotskyism is the theory of Marxism as advocated by Leon Trotsky. Trotsky considered himself an orthodox Marxist and Bolshevik-Leninist, arguing for the establishment of a vanguard party of the working class. His politics differed sharply from those of Stalinism, most prominently by opposing the policy of Socialism in One Country which he argued was a break with proletarian internationalism, and in his belief in what he argued was a more authentic dictatorship of the proletariat based on working-class self-emancipation and mass democracy, rather than the unaccountable bureaucracy he saw as having developed after Lenin's death.
Lenin and Trotsky were close both ideologically and personally during the Russian Revolution of 1917 and its aftermath, and some call Trotsky its "co-leader". Trotsky was also the paramount leader of the Soviet Red Army both before and in the direct aftermath of the Revolutionary period. However, Lenin had numerous rather biting criticisms of Trotsky's ideas and intra-Party political habits, including before the Revolution. In a 1914 article titled “Disruption of Unity”, Lenin struck at what he termed Trotsky's factionalist tendencies stating, "Under cover of ‘non-factionalism’ Trotsky is championing the interests of a group abroad which particularly lacks definite principles and has no basis in the working-class movement in Russia. All that glitters is not gold. There is much glitter and sound in Trotsky’s phrases, but they are meaningless." 
However, after Trotsky accepted that unity between the Mensheviks and Bolsheviks was impossible and joined the Bolsheviks (playing a leading role alongside Lenin in the revolution), Lenin said: "Trotsky long ago said that unification is impossible. Trotsky understood this and from that time on there has been no better Bolshevik" 
Trotsky's Fourth International was established in France in 1938 when Trotskyists argued that the Comintern (or Third International) had become irretrievably "lost to Stalinism" and therefore incapable of leading the international working class to political power. Today, in the English language an advocate of Trotsky's ideas is usually called a "Trotskyist" whilst Trotskyism's opponents usually refer to them pejoratively as a "Trotskyite" or "Trot".
- Lenin and Trotsky were "co-leaders" of the 1917 Russian Revolution: http://www.icl-fi.org/english/wv/archives/oldsite/2004/RCP-823.htm
- V.I. Lenin, "Disruption of Unity", 1914
- "Minutes of the Petrograd Committee of the Bolshevik Party," 1 November 1917
- The Transitional Program. Retrieved November 5, 2008.
- Collins Dictionary and Thesaurus (1993)