Here are some background links. The Wikipedia entry is long on analysis, but short on biographical information. This site focuses more on biography, and includes the nice feature of having a gallery of his paintings arranged by year. A short biography can be found here.
The older I get, the less trust I put in theories that have to do with humans and their acts. So it seems for art. To me, art is something of a craft, so I have no problem with rules-of-thumb such as "fat over lean" when painting in oils. That sort of thing isn't theory: it's a matter of practice that can be ignored if the painter so chooses. On the other hand, Kandinsky who apparently was always fascinated by color, applied his intellectual skills to the matter of painting, and before many years elapsed concluded that abstraction was what the facts (as he saw them) demanded. He created some of the first-ever abstract paintings a few years before the Great War, and by the war's end he had essentially abandoned representational painting.
Kandinsky is perhaps best known for paintings with lines and geometrical forms distributed on flat backgrounds. But he did more than that. Over his abstract art (or Non-Objective Art -- as the Museum of Modern Art called in in the 1930s) career, he did a good deal of experimentation. Often he would be playing with several themes simultaneously. In the Gallery below I present part of what he was working on around the year 1940. Also included are works from more distant years using elements of what he was doing around 1940. Kandinsky therefore can be seen to have added and dropped abstract painting styles over his career while usually playing with more than one ideas at any given time -- an evolving set of parallel projects or (for him) intellectual investigations.
First, a few paintings he made during 1939-1941. Note the variety.
Next, some paintings using layered or stacked Egyptian-like elements. He did these from time to time over a decade.
Besides geometric forms, Kandinsky also played around with organic blobs. "Fingered" blobs reappear in the paintings below made over another ten-year period.
One of his last paintings.