The term "abstract art" is like the term "modern music" in the sense that it is a very broad umbrella sheltering a wide variety of art. But like "abstract math," the general sense of the term is that it is the opposite of the concrete, or "realism." At one end of the continuum is a painting of a violin so perfectly rendered that we feel we could reach into the frame, pick up the instrument, and play it. At the other end is a canvas painted pure white or black all over. There is nothing in it to reach in and touch.
A simple, common definition of "abstract art" is "not realistic." Yet many artists who call their work abstract, actually do have a subject in mind when they paint. They take a figure or landscape and simplify it, exaggerate it, or stylize it in some way. They are not trying to imitate nature, but to use nature as a starting off point. Color, line, and form are more important to them than the details of the actual subject matter. They want to give a sense or feel for the subject rather than an exact replication.
Historically, the term "abstract" has been associated with a variety of art movements. The cubism of Picasso, Braque and Cezanne was a geometrical abstraction. In the United States , a group also known as the New York school of action painters were defined by critics as "abstract expressionists." Yet the individuals in this group varied greatly in their approaches. Jackson Pollock did overall drip paintings. Mark Rothko painted shimmering color field canvases based on a simple square pattern. Willem de Kooning did not abandon subject matter like the others, but abstracted the female figure in much of his work.
Art that has no intentional beginnings in any subject matter is sometimes referred to as "non-objective," or "non-representational." A related term is "minimalism," or the tendency to take as much away from the painterly surface of the canvas as possible. A white square painted on a white background is an example of minimalism. The end result is not so much the point as the daring it took to get there.
"Modern art" is another term commonly used to refer to abstract art, though originally this term was used to differentiate the experimenters of the twentieth century from the traditional European painters and sculptors. Thus, "modern art" began over eighty years ago, and is no longer new. Many movements in art have come and gone since then. For example, "pop art" incorporates popular culture such as comics and movie stars. Well-known artists of this genre include Andy Warhol, who painted Cambell's soup cans and portraits of Marilyn Monroe; and Jasper Johns, who did a series of flag paintings.
"Contemporary art" is another one of those terms that covers a wide variety of art. The best definition of "contemporary" is the work of any living artist, though the term has also been used to mean art that you would hang in a contemporary home. This sense of contemporary is more like the term "modern," in that it means the opposite of "traditional." Thus, "contemporary art" is also sometimes used to mean "abstract art."
Another way to define the term "abstract art" is to enter it as a search term on Google or Yahoo and look at the results. There will be millions of them, proving that the term is used today to cover a vast amount of art. I use the term "abstract art" to define my own painting because I know that people who love my art tend to define it this way. They often find me by entering the term on Google. Others use the term "modern art" or "contemporary art" to find me.
So where does that leave us in our definition of abstract art? Like most definitions of art movements, the answer is complex. We can look at it historically from an art critic's perspective, or use it as the general public would, to mean something other than traditional realistic representation.