By its very nature, defining what an abstract really is can be a source of much philosophical discussion. In the broadest sense, an abstract is something that is symbolic and does not represent reality. In photographic terms, then, an abstract image is something that does not reflect reality.
This does create something of a problem because everything around us is real (hopefully), so any image contains some element of reality; therefore it can’t be abstract by definition. Let’s cut through all the nonsense and be a bit more logical about it.
Perhaps an easier way to think about abstract images is to consider them as small slices of reality that, when viewed in isolation, give no clue as to what they are in real life. A very extreme example of this could be an electron microscope image of human skin at enormous magnification. Without some frame of reference, you would have no idea what you were looking at, so you could consider it an abstract image.
OK, as mentioned, that is a very extreme example. The truth is, though, opportunities to explore abstraction are all around us. You just need to know where to look.
In many cases, colour can also be a key identifier as to what the image is. Grass is green and sky is blue are obvious colour references that we understand. Remove colour from an abstract image and you take away another visual clue by which you could possibly identify it. You are left only with shape, tone, form, pattern and texture.
This is a positive boon to the abstract photographer because that is exactly what you are after. Another positive aspect of the abstract is that there really are no rules apart from getting your camera settings dialled in correctly.
Subject matter is no problem at all. There are so many abstract subjects you can cover in the world of architecture, both rural and urban. The world of the organic, flora and fauna, have huge possibilities, as does the realm of the inorganic. There is shape, form, pattern and texture wherever you look, enough to boggle the mind.
There are abstract opportunities in the shapes and repeating patterns of the man-made environment, as well as the more fractal patterns in nature. A good macro lens is a superb method by which you can isolate patterns at a small scale; and with the additional use of very shallow depth of field, you have even more scope to create arresting and thought-provoking imagery.
There is also much abstract imagery to be had at a larger scale as well. If you happen to live in a large modern city, there are no doubt a great many glass-fronted buildings where the surfaces create many reflections and patterns within their structures.
The bottom line is that if you can shoot an image that contains interesting shapes, patterns and tones, but you cannot easily recognise what the object might be, then you have your abstract.