Leninism, Stalinism and Trotskyism In A Nutshell

Of the three well known Russian Revolutionaries, Leon Trotsky is probably the least well known apart from the fact that he was name checked in a Strangler song and was killed by an ice pick.
Now that Tony Blair has come out as a former Trotskyite, interest has grown in the man and what he stood for and how it differed from Marxism explained here.
Fact is, Marxism is the idea put forward by Karl Marx of a community (hence the name Communism) which is ruled by the people for the benefit of all while Leninism, Stalinism and Trotskyism are the interpretations of the ways of achieving Marxism. 
A quick review of the Russian Revolution is required so we have Lenin, Stalin and Trotksy at the head of the Bolshevik Party who disposed the Royals and found themselves in charge and wondering what to do next. 
Lenin was the boss and he, Stalin and Trotsky decided what was required in order to achieve Communism would be the need to set up a ruling Party, named the Vanguard, to oversee the installation of Communism and which would then fade away in time to leave the workers in charge.
So far, everybody on the same Marxism page but now comes the difference because Marx said for Communism to succeed, it would need to spread around the capitalist nations of the World but Lenin wanted to ensure Communism was settled in Russia before spreading the ideal out to other countries, this is called Permanent Revolution as in a domino effect where there would always been one country following another into Communism.
Trotsky didn't want to wait and wanted to strike while the iron was hot and spread it far and wide while Stalin wanted to keep it only in Russia and to turn the vanguard party into a permanent Government.
Lenin and Trotsky therefore were together for the Permanent Revolution but Stalin was against the idea but they managed to keep the far more authoritarian Stalin subdued until Lenin died in 1924 and Stalin grabbed power, exiled Trotsky to Kazakhstan and executed almost all of the Party's Trotskyists in what become known as the Great Purges and effectively removed all of Trotsky's influence.
Stalinism took over and to stop Trotsky's continued criticism of Stalin's rule, after several assassination attempts, he succeeded and he died in his Mexican home from a pick axe to the head in 1940.
Trotskyism then is the idea by Leon Trotsky that the Marxist theory should be spread around the World rather than only one state.