It's to do with how the actual memory is organised inside each chip on the DIMM. if you think of memory as a number of cells organised into a certain number of columns, like a grid, then x16 refers to the number of columns.
The organisation is usually marked on the chips themselves. For example, you'll see things like "128M8" which means 128MBit x8, so that DIMM can be said to have x8 organisation.
Double sided is a bit of a misnomer because it's actually the number of memory banks on the DIMM that's important - memory on the DIMM can be organised into either one or two banks. Usually, if there are chips soldered to both sides of the DIMM then it's a double banked (wrongly called "double sided") DIMM, and if there are chips soldered to only one side then it's single banked. But there are exceptions to that rule.
Anyway, the upshot is that if there are chips soldered to both sides of the DIMM, AND the organisation is shown as x16, then it probably will not work on that motherboard. If there are chips soldered to only one side of the DIMM, OR the organisation is shown as x8 or x4, then it probably will work.
However, as I said, there are exceptions to that rule because the side of the DIMM the chips are soldered to does not necessarily correspond with the number of memory banks the DIMM supports.