Watch this documentary by Simon Schama, part of the Power of Art series. It's beautiful and worthwhile. Try to see it in the highest quality possible.
And if you get a chance, read Rothko's own little manifesto. It's a fascinating little book (which he never intended to be published) about his views. Rothko spent a lot of time thinking about what art is and isn't, and the role of the artist in society.
He said: "A painting is a statement of the artist's notions of reality in the terms of plastic speech."
The Artist is not a technician, for skill is just a sleight of hand. The advertisement illustrator, however skilled he may be, makes pictures, but not art -- his primary mentality and motivation is actually that of a salesman. The artist, as Rothko defines it, is a philosopher. His work expresses a perspective on the world that cannot be described in words, because he is inventing language. The stronger the artist, the more distinctive his language (also known as "style"). The artist, like the philosopher, generalizes beyond the particular, though the plastic elements of art find many other employments in society: decoration, illustrations, informative graphics, road signs, etc. Mathematicians, similarly, apply their craft to settling accounts, puzzles, computing, gadgeteering and engineering vacuum cleaners. But the real function of these two characters is to evolve generalizations.