Why neural nets are split into layers?
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.