Why is every artificial neural network layered? Why isn't each node just a separate process?
A neural net will work with any arbitrary topology, as long as it has no cycles. We often use layers because it is easy to implement and because it allows a human to set parameters for how large the net should be, such as "four layers with 5, 6, 7 and 10 nodes, respectively".
I cannot find anything on the web about neural networks that are not organised into layers, but there is nothing principally impossible about them. Here's an idea for a rainy sunday afternoon: implement a neural net as a directed acyclic graph and use an evolutionary algorithm to find a good topology. Compare it to a layered neural net with similar numbers of nodes and edges.
There are some hand wavy intuitions that adding depth (stacked layers) allows the formation and combination of higher level features. So you can take features from the layer before and build more complex ones by combining them together, rather than starting from scratch.
Conv nets are the poster child for this. You can see that their low level layers learn things like gabor filters and the deeper layer learn things like face detectors.