Malevich unveiled the Black Square in 1915 as the first example of a new style of art he dubbed Suprematism. As was characteristic of Russian avant-garde artists in the early 20th century, he wrote numerous manifestos to give a philosophical basis for his art. In these, he argued that art should be a reflection or abstration of thoughts, feelings, and ideas, rather than of physical objects. To achieve this, he made a complete break with representational art and began with as simple a composition as he could conceive: a single shape with black and white as the only colors. Thereafter his works became somewhat more complex, though still following the same theme. Amongst his many other Suprematist works (most of which were titled simply Suprematism), some of note are: Suprematism (with Blue Triangle and Black Rectangle) (1915), Suprematism (with Eight Red Rectangles) (1915), Red Square (1915), Suprematism (1917-18), and Suprematism (1921-27).
The Russian Avant-Garde
Suprematism was heavily influenced by (and itself influenced) several other avant-garde movements of the period. Constructivism was just beginning to take hold at the time; this philosophy (simplified immensely) advocates focusing on fundamental building blocks of artistic works rather than coherent wholes. Thus Constructivist poetry, which became quite popular during the 1920s in Russia, viewed words as entities in themselves (important qualities include how they look when written, how they sound out loud, and so on) rather than merely as placeholders for their definitions, and thus poems as collections of these building blocks. Following a similar vein, Suprematism focuses on shapes and colors in themselves, rather than using them solely to depict more familiar objects.
Cubism was also enormously popular at the time, and Malevich himself had done some Cubist paintings. Clearly the geometric shapes of Suprematism bear a striking resemblance to Cubism, and the overall aesthetic sense probably owes quite a bit to the Cubists, but the process has been changed in an important respect: whereas Cubism takes a physical object and breaks it down, sometimes so it is no longer recognizable, Suprematism takes an idea or feeling and build up an artistic representation of it from simple forms and colors.
The 1917 Revolution
The Russian avant-garde, with its view of itself as a revolutionary and ultra-modern movement, formed a natural fit in many ways with the Bolsheviks behind the 1917 Russian revolution. Many of their manifestos, such as the 1913 "Slap in the Face of Public Taste", are even similar to Communist manifestos in writing style, and their rejection of previous styles of art meshed well with the Bolsheviks' distrust of anything dating to much before the revolution as potentially bourgeois or even Tsarist.
Along with the Futurists, who based their view of art on an almost fanatical love for the aesthetics of technology (which in the early 20th century meant being enamored with noise, machinery, smoke, and all the other trappings of a nascent industrial society), they would become the artists of choice in 1920s Russia. Malevich himself displayed little interest in politics, but saw the building of a new social order as a good opportunity for his artistic vision, and drew up gradiose plans of entire cities organized on Suprematist principles (which were of course never realized).
However, the rise of Stalin to power led to the destruction of avant-garde experimentation beginning in the late 1920s. After abandoning his Suprematist style for more Cubist works through the 1920s, he created a final version of the Black Square (the fourth in total) in 1929. It is similar to the first, but the square is much more uniformly pure black and painted with thicker coats of paint. Thereafter he tried his hand at more realistic representational works to satisfy the dictates of Stalin's Socialist Realism, but fell out of favor and died in poverty in 1935.
This Is Art?
Of course, the obvious reply to all this is "well, but it's still just a black square on a canvas; how is that art?" That's a question a bit beyond the scope of this article, but in any case, a trip to any modern art museum should make clear that Malevich's Black Square has at the very least been enormously influential.
The links to images of Malevich's works here all come from the Malevich page at the Russian Avant-Garde Gallery, which has an extensive chronological listing of many of his other works (also with images).
The Artchive page on Malevich has some additional images.
Some class project at USC has a page with images of a few assorted Suprematist works from Malevich and others.
Mayakovsky.com has a ton of information on the Constructivist and Futurist poet (one of the authors of the "Slap in the Face of Public Taste" mentioned earlier), including some of his poetry.
This is the third in a series focusing on particular important works of art in their historical and artistic context (it used to be called "Art Piece of the Week," but the new name is a bit more accurate). The previous two if you missed them: Cycladic Lyre Player and Magritte's L'Empire des lumičres.