Trotskyism is the theory of communism as advocated by Leon Trotsky, or is more loosely used to denote various political currents claiming a tradition of Marxist opposition to both Stalinism and capitalism.
Trotsky advocated Proletarian Revolution as set out in his theory of "Permanent Revolution", and he argued that in countries where the bourgeois democratic revolution had not triumphed already (in other words, in places that had not yet implemented a capitalist democracy, such as Russia before 1917), it was necessary that the proletariat carry out the tasks of that revolution and make it permanent by carrying out the tasks of the social revolution (the "socialist" or "communist" revolution) at the same time, in an uninterrupted process. This theory was advanced in opposition to the position held by the Stalinist faction within the Communist Party that "socialism in one country" could be built in Russia.
On the political spectrum of Marxism, Trotskyism is considered to be on the left. This characterization reflects their insistence on principles such as internationalism. Expressed in derogatory language, they are described by their ideological opponents as "left deviationists" ("levye uklonisty", in Russian).
Trotsky later developed the theory that the Russian workers' state had become a "bureaucratically degenerated workers' state". (The similar Eastern European communist governments which came into being after World War II without a revolution were later referred to as "deformed workers' states" by some Trotskyists.) Many of Trotsky's criticisms of Stalinism were described in his book, The Revolution Betrayed (http://www.marxists.org/archive/trotsky/works/1936-rev/index.htm).
"Trotskyist" has become a by-word used by Stalinists to mean a traitor; in the Spanish Civil War, being called a "Trotskyist" or "Trotskyite" by the USSR-supported elements implied that the person was some sort of a fascist spy or agent provocateur. George Orwell wrote about this practice in his book Homage to Catalonia and in his essay Spilling the Spanish Beans. Note: while both "Trotskyist" and "Trotskyite" are words that were probably originally coined by Stalinists to mark those who sided with Trotsky in factional disputes, "Trotskyist" is a term that is claimed by many, though not all, adherents of Trotsky's views. In fact the term Trotskyist was originally used by the Stalinists to describe that faction of the Communist party, led by Leon Trotsky, that described itself as Bolshevik Leninist (although not used with any great frequency today, this original term is still used by some). However, "Trotskyite" retains its pejorative connotation, and in some circles, it (or "Trot") is used indiscriminately to refer to any far-left adherents derisively.
In 1938 Trotsky established the Fourth International. After his death this organisation has split many times. Trotskyist parties and groups are notorious for their tendency to split into smaller groups, quarrelling over theoretical differences that seem insignificant or indecipherable to an outsider, but which sometimes have major practical consequences for those who hold those positions.
Many developed countries have several different organizations which claim some tendency of Trotskyism. Among the largest Trotskyist organisations today are the Socialist Workers Party and the Socialist Party (formerly Militant) in Britain, Lutte Ouvrière and the Ligue communiste revolutionnaire in France, the International Socialist Organization in the United States, and the Partido Obrero in Argentina. Three smaller Trotskyist groups are the Spartacist League (see International Spartacist Tendency), the League for the Fifth International (in Britain consisting of Workers Power) and Socialist Alternative in Canada.