So many people must be asking this question all the time ... and not getting satifactory answers because I keep seeing articles on the subject.
So, to add to the mix, I thought that, because I am an Abstract Artist, I would give my answer.
First lets see what the official version has to say. The second definition of "abstract" in the Concise Oxford Dictionary says "Idealistic, not practical; abstruse; (Art etc.) free from representational qualities" And it is this last description I want to look at.
That which is free from representational qualities is a picture (or other type of art discipline) that does not depict any recognisable image such as a figure, building, or sky. There is no purposeful reference to anything physically particular. Jackson Pollock, for instance, was one of the first abstract artists and produced the finest of examples... take a look at this: www.nga.gov/feature/pollock/painting.html
If you study this image closely it will soon become apparent that he randomly dripped paint onto a board laying flat on the floor. And if you read up a bit about him you will find that he even suspended paint-filled cans above a board, punched holes in the bottom of the can, and allowed the cans to swing, or be nudged, the paint slowly dripping to give a totally no-representative image. The important thing to remember here is that he had no intention of producing any kind likeness to anything physical whatsoever. So this style of art is truly an abstract work.
The difficulty comes when a piece of art is produced in a very loose and, even, child like way. Some might regard it as abstract. Here is an example of what I believe is NOT abstract art: go to www.guzzardi.it/arte/ then on the left side click on "Artisti Ceris" then click "D" go to the fourth name from the bottom "Jean Dubuffet" and click "Art Brut" there you will find three images of his work ... each one depicts roughly painted figures and faces, and although the left hand image looks like imageless lines I do believe there are faces to be seen there (Don't get me wrong - Jean Dubuffet produced many abstract works ... in fact that is what he was most famous for ... it's just that these examples are not).
Some might call these Dubuffet paintings semi-abstract - but I do not. For me there is no such thing - either the picture does not represent anything ... or it does - no matter how basic.
Another puzzle might be the later works of Mark Rothko. Look here: http://www.nga.gov/feature/rothko/late7.shtm Some might argue that a basic rectangular shape could be regarded as a representational image. But you need to read about his life and philosophy to understand that he was not painting rectangles but what has become known as colour fields. The scale of his massive works and the fact that he gave up even titling his pieces should dispell any possibility of his work being anything other than the purer form of abstract art.
I would like to finish with one of my own enigmatic examples: www.ablot.com/passion3.htm in which I have sort to "portray" the idea of deep or spiritual movement. Such an explanation is, of course, dangerous in that I have used words like "portray", "spiritual" and "movement" - all of these have associations ... but not necessarily physical ones! It is my belief that there is a relationship to be had with my spirit ... and it is the purist form of abstraction. You can find out more about that in my blog on my website.
The only other thing to say here is that the abstract artist is seeking to put on canvas such non-figurative "imagery" as emotion, thought, the greater self, anarchy, and total randomness ... to name but a few. And I intend to examine these "non-figures" in a bit more detail in my articles starting with "How to produce an abstract image - Part 2" You will be very welcome to come along and read it!