Well, that's where nets come in. Errr, this is hard to explain. A net is a connection, in its basest form. Imagine two components in series, say an RC circuit. Then there is at least one net in this circuit--the wire between the R and the C comprise one unique net, and the wires on the other sides of the R and C should also be separate nets of their own. Physically, I am guessing an entire net would remain at the same potential, since there would be potential drops across either the R or C (although I never really asked).
Hence the usage of planes in layers. A plane is basically a giant net throughout the layer. So, one of the inner layers of the board is a power layer, and the net assigned to it might be -5VA or +5VD. In other words, any through hole connected to the layer will have a constant supply of power at that potential, and thus all the components connected to that can interact with each other.
Anyway, enough technical talk. I'm not sure if I can explain it any better anyway. It's not something you can usually tell just by looking at a board either, so it's probably not one of the more obvious things. On the other hand, I am proud to present, after another week's worth, the routed (more or less) TDC board. Too bad I'm still not done with it.

If you are curious, all the red parts and green traces are on the top layer, and all the blue parts and traces are on the bottom. So it doesn't matter that red overlaps blue and blue overlaps green. What DOES matter are the positions of the through holes. It was often frustrating to find that a via on one side would hit a component on the other side. Then I'd have to move the via or the trace, which sometimes led to moving other traces, etc. It was bad enough that everything was so condensed.....
Thank you for dumbing it down but it's still hard to follow. Like Herodotus.
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