Frollard is quite right. Another complication for multi-layer boards is accessibility. With a single layer board, the conductive paths ("traces") are on the surface. If there is an error on the board you can go in with a knife to cut the trace and you can solder wire to the surface traces to patch it to another location. With a multi-layer board cut and patch is much harder. Combine that with the cost and you have to be sure to get the multi-layer board right on the first go around. You can think of the multi-layer board like some complex highway interchanges called stack interchanges (here is a Wikipedia link with lots of pictures). If you have a couple highways coming together you have to stack bridges up one on the other so that everything can pass through the area.
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One has a single layer of conductive material...others have multiple layers of conductive material (almost always copper) with insulation between them, and paths called 'vias' that allow different layers to conduct to each other.Commonly in cheaper electronics you'll find single or double layer boards, with expensive high end stuff (like cel phones or computer motherboards) using upwards of 9+ layers. Modern electronics are VERY complex and maintaining thousands of connections between various components without having them cross over each other is very difficult in a single layer; so they split up the paths on multiple layers.