This is usually a bit out of my comfort zone, but I read a bit on communist history in the past few months, so I'll give it a shot!

You are asking about various forms of communist thought here, four of which - Maoism, Stalinism, Trotzksyism and Leninism - all build upon the ideas of Karl Marx as formulated in the Communist Manifesto (Das Manifest der Kommunistischen Partei, 1848) and The Capital (Das Kapital. Kritik der politischen Ökonomie, 1867/1885/1894 [published in three volumes]), both books written by Karl Marx and, in the case of the manifesto, assisted by Friedrich Engels. In the case of the Capital, Volume 1 was written by Karl Marx and Volumes 2 and 3 were written by Engels following what Engels claimed to be notes of Marx. So it is interesting to note that only a third of one book and half of the authorship of the other actually go to Karl Marx, with the rest being the work of Friedrich Engels.

Now, marxism was largely theoretical in its early shape, with Marx's works mostly consisting of his observation of the working class cultures in Germany and Britain during the 1830s through the 1890s.

This already brings us to the first problem with marxism that presented itself to the Russian bolsheviks and the Chinese communists (they wouldn't develop 毛泽东思想, the Mao Zedong Thought, more popular as Maoism in the west, until after the founding of the People's Republic of China; so referring to them as maoists at that time might be inaccurate): Germany, Britain, France, Belgium were the places that Karl Marx knew and in which he studied the "proletariat", the working class that in the marxist worldview is the final oppressed class destined to overthrow the ruling class to eventually create a global egalitarian society.

Neither China nor Russia showed the industrialization or urbanization levels of Western Europe at the time. Karl Marx almost exclusively considered industrial workers in his thought, which was appropriate considering his and Engel's backgrounds; but Russia and especially China had far more farmhands than they did assembly line workers.

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So let's first consider the position of Vladimir Lenin and his movement in the Russian Empire.

Marx is dead since 1883, Engels since 1895, it is now 1917 and your country is in total ruin after three years against a logistically and technologically superior enemy. The public belief in the old authoritarian "divinely guided" tsarist system is largely shattered, starvation runs rampant, the families want their men back from the front in order to regain some form of secure income. Add to that the fact that the old elite really doesn't want you to take power and overthrow the old religious and social structures - next to all the starving children and dead soldiers, you also have a civil war on your hands.

So Vladimir Lenin is forced to be far more pragmatic than one might think when considering he is the founder of the Soviet system and the centerpiece of so much of contemporary communist idealization.

This leads Lenin to a problem: Marxism in its purist approach is rather constrained, as Karl Marx was rather precise in what he believed to be necessary to set up the global communist society - and among these things was a modernized, appeased, largely industrialized society. Marx had anticipated Germany, France or Britain to be the birthplace of the red revolution; never would he guessed that it would be the backwater that was Russia at the time.

This brings us to "Leninism", the unique form of communist flavor added by Vladimir Lenin. This is not to be confused with "Marxism-Leninism", a term coined under Stalin to describe the ruling system in the USSR.

The main part added by Lenin is the concept of the "Vanguard Party" to safeguard the revolution. Marxism in itself was not necessarily the blueprint for the brutal totalitarianism that we see in the Soviet Union, the PRC or Kampuchea, to name a few example. I would argue that this totalitarian note was primarily added through this very concept, the Vanguard Party. Marxism had originally envisioned a societal ideal that one might liken more to the views of contemporary anarchists, whereas Leninism clearly endorsed the all-powerful One-Party-State, with no division of power and the unification of all political force in the communist leadership for the duration of the socialist transition. Hence, "Dictatorship of the Proletariat" as envisioned in Marxism becomes replaced with the Vanguard Party concept and the One Party State of Leninism (note that Leninists do view these concepts as non-exlusive, so they called back to both of them).

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This brings us to Trotskyism and Stalinism, both being spinoffs of Leninism.

After Lenin's death in January of 1924, it was not immediately apparant that a power struggle would break out. Leo Trotzky had been a successful military leader and political speaker for the Bolshevik cause, whereas Joseph Stalin was rather a regional figure, mainly in the city of Tsaritsyn, a city that he would later rename due to that occasion and that is better known to us as Stalingrad.

The difference between Trotzky and Stalin philosophically was mainly a split in diplomatic philosophy, with the concepts of "Permanent Revolution" (Trotzky) being pitted against "Socialism in One Country" (Stalin).

Basically, do we need to stabilize first and then act carefully or should we push for the global revolution immediately and permanently?. With Stalin's rise to secretary-general of the party, Trotzky's expulsion from the CPSU in 1927 and his exile from the USSR in 1929 as well as the eventual assassination of most Trotzkyists in the Great Purge of 36 to 38 and Trotzky himself in 1940, it is clear which thought won out.

Note that "Stalinism" is not only a philosophical term referring to a certain interpretation of marxist thought, but also a politohistorical one referring to the USSR during Stalin's time as a whole. That is why Stalinism is a term so associated with totalitarianism, murder of innocents, genocide and extreme foreign aggression. One should not make the mistake to anticipate Trotzky's plans for the Soviet Union to have been any different, especially considering that "extreme foreign aggression" was kind of his thing. Stalinism is furthermore notable for the introduction of nationalism into the ideology, with Stalin, an ethnic Georgian himself, having written about nationality in Marxism and the National Question as early as 1913. A visible manifestation of nationalism within communism is the puppet state system established by Stalin after World War 2. He introduced Estonia, Latvia and Lithuania into the Soviet Union directly, but let Poland, Hungary, Czechoslovakia and the others maintain a veil of independence through being formally separate countries. That would have not been as easily possible in Marxism, Leninism and Trotzkysim, all of whom view national borders and boundaries as unnecessary and outdated and advocate the creation of a single "nation" spreading all humans and the entire globe. By the way, this concept of nationalism has been driven to the extreme under Juche, the system of stalinist communism embraced by North Korea, a country so hyper-nationalistic that there are some very good arguments to classify it as fascistic rather than communist.

Sometimes, the ideological side of Stalinism is referred to as "Marxism-Leninism" for the sake of clarity, but I personally oppose this term, as it was chosen by Stalin himself to give his ideas greater legitimacy.

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Alright, almost done. Now, for Maoism. Mao during the Chinese Civil War had not been that much of a political thinker, rather taking on the mantle of the acting commander. He generally followed stalinist ideals, seeing the nationalistic aspect of it as a good way to preserve his position of power against any theoretically possible attempt by the Soviets to form some sort of Sino-Soviet super union.

However, Stalin died in 1953, and his successor Khrushchev was an Ukrainian and was thus pretty displeased at what Stalinism had done to Ukraine in particular. As the Soviet Union stopped the Stalin Cult, Mao saw his chance to develop a unique branch of communism in what became known as the Sino-Soviet Split in 1956. China openly positioned itself against the Eastern Bloc and the Warsaw Pact and sought friendship with the non-aligned nations of the world. China, seeking to gain safety from both American and Soviet invasion, developed nuclear weapons in 1964 and conducted its first thermonuclear test in 1967.

Maoism (Chinese: Mao Zedong Thought, 毛澤東思想) shifted the focus of the marxist fixation on industrial laborers to the rural peasantry, of which China had far greater numbers, as I mentioned before. In 1964, Maoism was formalized in the Quotations from Chairman Mao Tse-tung (毛主席语录), more popularly called the Little Red Book. In it, Mao importantly deals with perceived or real foreign or domestic enemies (among them the Kuomintang, the Japanese, the Americans and the Burgeoisie), so there certainly is a great amount of nationalistic strife in Maoism as well.

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Alright, as an overview (and looking back on it, even a timeline), that should be a fine-enough picture of various communist iterations. Other notable schools of thought might include Hoxha, Tito, Ho Chi Minh and the Eurocommunists of the Cold War, but you did not ask about them specifically, so I will leave it at that.