The Wikipedia entry on Abstract art (link) starts with the sentence: Abstract art uses a visual language of form, color and line to create a composition which may exist with a degree of independence from visual references in the world. We will see the first part come back in the subsequent themed contributions, but it is also important to pay attention to the second part: the presence of identifiable items do not make an image a non--abstract. I stress this, because I have often encountered comments that assume abstracts by definition cannot have identifiable elements.
Abstract art is mostly encountered in paintings, and less frequently in photography. It is worthwhile pointing out a big distinction between these two abstract art forms. Painters can let their imagination go wild in creating their abstracts, with forms, colours and lines fully unrestricted from a practical point of view. Photographers are limited to what they encounter (or in rare cases set-up) in real life. It is essential to develop a good eye for possibilities in this respect, keeping in mind that one can select larger scenes as well as small details (without necessarily going into the special field of macro photography).
Let's end this introductory piece with a few general remarks. First, as in all photography, composition plays an important role in determining the quality of an abstract shot. In my experience the natural flow through an image, often enhanced by good choice of using the diagonals, is crucial in this respect. Second, always shoot a series of pictures of one subject, so you can choose the best one. Third, and one that I wished I had known earlier, if your abstract is a detail shot, also take a shot of the overall subject for reference. It is embarrassing to have a successful abstract photograph, and when asked for more information, failing to remember what it actually was.