Basically, anything you can compute using the information in the image qualifies as a "feature" of the image. Average brightness? It's a feature. Third moment of the Knight's Tour of the image? It's a feature. MD-5 hash of the four corner pixels? It's a feature.
How many features does an image have? Answer: an infinite number.
How many useful features does an image have? No-one knows, and the number is "unknowable" at this time. Quite possibly infinite.
Which features should you use for your purposes? You need to experiment. I would not advise looking for the "best" set of features, by the way: a better feature set is likely to be found within a few hundred years at most.
You'd have to use image processing. So that answers the first question.
For the second question, I don't use neural networks so I can't answer that question with as much authority, but I guess you'd use the Neural Network Toolbox to create a neural network, and send your image into it.