Modern Art is an official designation, and in specification of this we capitalize both words, as opposed to modern art, which would refer to art containing elements of modernity, Modernism, or historical relation to the Modern Period of Western history. Modern Art, however, is a specific art movement which spanned largely the post-WWII Western world and continued on into the 1950s and '60s. This style included many different sub genres, which I'll go through. On the whole, however, the individual sub genres collect into the overarching label of Modern Art as art which reflected the dissemination of the earlier Modernist period.
Modernism, you'll remember, spans back to Manet and Baudelaire, two artists who saw the need to reevaluate the world in the context of Industrialism and an increasingly global environment (at that time visible through the institution of British Imperialism). As the Western world moved into the 20th century, technological revolutions led to a form of hyper-industrialization and, in a way, the reinvention of a new world. By the time of the First World War, it was an unrecognizable place as compared to the conditions of a hundred years prior. The Modernists responded to this new uncertainty about the world, and the most central expression of this Modernistic sentiment was Impressionism. Monet's style of impressionistic brushwork was a new approach and became in fact the definitive style for the new era. This was Modernism.
Postmodernism is something different. Like the expansive exploration of the Post-Impressionists to new experiments in theory and form, Postmodernism takes Modernism to a broader extreme. In literary theory we are taught that Postmodern literature is, essentially, Modern literature taken further—and by "taken further," I mean not only expanded theoretically or practically but culturally, globally, and ultimately ideologically. The artist's craft is changing forms, but the artist's perspective on life and art is also changing. In a way, the modern world had fully established itself by the end of the Second World War, and artists during this time existed in an artistic period which was new but also reflective of the unexpressed, unfulfilled aspects of Modernism that budded out briefly in the Post-Impressionistic and Abstract periods of the previous generations. The groundwork had been laid; new applications would be forged through that. (Postmodernism famously takes from other sources as a characteristic trait, like how Duchamp "stole from" the Mona Lisa for his L.H.O.O.Q.). This kind of expanded, fully-realized Modernism (what we call Postmodernism) provided the most direct inspiration for artists during the Modern Art period.