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Communism = NOT a system, a state, a type of government, or an economy that one puts into place or forces others to put into place. Rather, communism (little c!) is a term used to describe the tendency in human history towards community. This is somewhat confused by the fact that we can also use "primitive communism" to describe specific tribal societies in which property was held in common yet the means of production were not sufficiently developed to produce complex global culture. Nevertheless, "communism" was used by Marx primarily to describe an ongoing historical tendency:
"Communism is for us not a state of affairs which is to be established, an ideal to which reality will have to adjust itself. We call communism the real movement which abolishes the present state of things. The conditions of this movement result from premises now in existence." - Karl Marx
Marxism = aka "Scientific Socialism" is the body of thought first developed by Karl Marx and Friedrich Engels. Unfortunately the top commenter has identified Marxism primarily as an "economic critique" of capitalism, which doing Marx a terrible disservice. Marxism is not merely an anti-capitalist theory: it is a fully worked-out scientific philosophy that engages with the material world on rational terms. It is literally the most powerful synthesis of human thought ever assembled, predating modern discoveries in everything from evolution to quantum mechanics, and that is what it needs to be since it aims to supplant the dominant ideology in society (bourgeois ideology; capitalism, the free market, etc.)
The core of Marxism can roughly be described in three parts: Dialectical Materialism, Historical Materialism, and the Labor Theory of Value. I'll attempt to describe them:
Dialectical Materialism is the philosophy of Marxism. It teaches us to look at the world so that rather than seeing things as abstracted concepts, we may appreciate them in all their life and movement. At the same time, Marx’s assertion that our consciousness is determined by the material conditions of our lives acts as a grounding anchor in communist political work.
Historical Materialism is the application of Marxism to the study of human history. Through this lens it becomes clear that the evolution of culture and civilization arises not from the great ideas of a chosen few, but rather as a direct consequence of the means by which the reproduction of society is organized.
The Labor Theory of Value is at the core of Marxist Economics. Although Marxism can hardly be reduced to an economic discipline, many have best come to understand it through these principles. For Marx, value is a social relationship, a living interaction between people. Understood as such, the most basic truth about capitalism is laid bare: that it fundamentally relies on the exploitation of human labor.
(please note that what the current top commenter is describing as Dialectical Materialism is actually Historical Materialism)
Of course, Marxism has been developed further since the 1800's and there are many strands out there. Guy Debord's Society of the Spectacle, for instance, has been called "the Das Kapital of the 20th Century" and forms the Marxist basis of the critiques of mass culture we see in films/books like Fight Club.
Leninism = We need to get something straight: much as Marx never called himself a Marxist, there was no "Leninism" until after Lenin was dead and the Soviet bureaucracy under Stalin needed to invent a state religion to justify its own existence (think of Juche in North Korea, except that "Leninism" was imported to Marxist organizations all over the world).
The basis of "Leninism" as we see it applied today is chiefly based on a caricature of the Bolshevik party as it existed in the darkest days of the Russian Civil War, i.e. at its most centralized, militarized, and authoritarian. In particular the type of "follow Moscow's lead" faux-internationalism that was imported to the global communist movement played a major role in destroying revolutions in Spain, Greece, France, Italy, etc. (and that's just in the first half of the last century!)
This caricature (Leninism) consists primarily of two ideas: firstly, that the working class in and of itself cannot reach revolutionary consciousness in the brief window of time offered by revolutionary situations caused by material conditions in the breakdown of capitalism, and so in order for socialism to prevail there must be an organized intervention by an intellectual class. I'll leave the debate there, but suffice to say that questions of leadership are very important in Marxist strategy.
The other pillar of "Leninism" is the so-called theory of imperialism, in which Lenin lays out his belief that in the "final stage" of capitalism, conflict will take place not so much between classes, as between "imperialist" states and non-imperialist states (see: colonialism, USA in South America, so on and so forth). This ideology has unfortunately led to all sorts of (IMHO) ridiculous and anti-Marxist politics, as the Marxist position has always been that the ruling and working classes are both international and as such resistance to capitalism should always be based along class, not national, lines.
Stalinism = after the defeat of the Western revolutions in the wake of the Russian revolution - in particular the German revolution of 1918, the soviet state was isolated and forced to survive in material conditions completely inhospitable to socialism. Socialism can only be international, since it relies on the idea that all of humanity will have common access to the latest technology and techniques. Russia was left with an embryonic workers' state without the sophistication and development to actually implement socialism (Russia had not fully developed its capitalist economy prior to the revolution).
As a result, a bureaucratic class arose to manage the state economy and dictate what would be produced and where, typically with very little emphasis on the production of consumer goods. This style of economic management and political authoritarianism is what is commonly known as "Stalinism". The ability to concentrate all state resources into the development of industry allowed for tremendous economic growth that has never been matched by any capitalist economy, allowing Russia to become a superpower almost overnight, but this type of state-managed capitalism has never been able to solve fundamental problems of the boom and bust cycle. And obviously, it has never created a true workers' state.
We should also mention that much as the Stalinists are the only ones who speak of "Marxism-Leninism", so too is it mostly the Trotskyists who critique anyone as "Stalinist".
Trotskyist = The top commenter's information here is pretty spot-on.
Maoism = There was a failed revolution in China in 1928, after which the defeated communist party fled to the countryside and established a base among the peasantry, who were the largest class in Chinese society at the time.
Over enough years, "Maoism" developed - the idea that the peasants could be the actual revolutionary class, and that power could be taken through a protracted "people's war" in the countryside, eventually capturing enough territory to surround the cities and take power. The ongoing civil war in India is probably the best present-day example of Maoism in action (see Nepal, as well).
How Maoism plays out in the first world is sort of too ridiculous to explore since its peasant-based ideology is turning Marxism on its head in the first place, but it will often center heavily around aesthetics and armed struggle of some kind. The cult of personality is always huge, and curiously a positive attitude towards Stalin is present since the Sino-Soviet Split (where Chinese and Soviet foreign policy began to clash) happened after Stalin's death. The Chinese dressed their interests up in the veil of "anti-imperialism" and a sort of "third world revolution" across Africa, Asia, and South America. A very handy strategy for gaining access to those markets and raw materials, of course.
This clash in foreign policy is also the reason why some people will speak of "Marxism-Leninism-Maoism" as opposed to "Marxism-Leninism": they represent the official state religions of Maoist China and Stalinist Russia respectively, each in competition with each other as a capitalist nation managed by a bureaucratic elite but competing in the world capitalist market according to its rules, and neither representing anything close to the true definition of Marxism or communism.