What is abstract painting?

To define abstract art is a bit like trying to define that which seeks not to be defined. You have to love artists, right? Here's one way to consider it:
An abstract painting seeks to distance itself from a strict visual representation of something you would typically see. This typically falls into one of two categories.
The first category involves a representation that is drastically simplified. Imagine a typical painting of a cat, which looks quite similar to any typical cat you could see. In an abstract painting of a cat, there may only be an outline of the cat. Perhaps it has a long tail and pointed ears. Maybe it has purple stripes and its body shape is a triangle. By examining it, you can tell that the basic figure at its simplest form is that of a cat, but the painting does not seek to strictly represent a cat. There is both a simplification of the cat's most basic qualities and perhaps some creative liberties taken as well in how it is represented.
The second category seeks to represent nothing physical. Instead of capturing images seen in a physical landscape, the painting uses shapes, colors, forms, and space to convey an emotion or an idea. Check out the painting Alpha-Phi by Morris Louis or Yellow Islands by Jackson Pollock for examples of this type of abstract painting.