A floating chain works very simply, by eliminating the derailleur. Of course, it predates the derailleur, so the above isn't quite correct.
You shift by hand. Doesn't matter whether the multiple cogs (or chainrings) are at the front or rear, but I never have seen both.
The chain length is such that it just allows shifting onto the biggest cog/chainring. In other combinations, it hangs loose. Very loose sometimes, which can cause problems.
The problem that can occur: the chain rides up on the rear cog, and skips. Same principle as when you ride a new chain on a worn freewheel.
The rear pulley on the dropout (as on the Integral, reportedly, this was invented by Velocio) works only if there is only one cog (with multiple chainrings). It prevents the chain from riding up, so it won't skip.
As detailed in "The Dancing Chain," early inventors of shifting systems had problems with taking up the chain slack. Various systems were used that kept the chain length constant (double chains, one freewheeling, the other engaged and similar ideas), until the spring-loaded derailleur came along.
The "floating chain" simply didn't bother to take up the chain slack. This made it possible to retrofit any old bike with multiple gears - simply stick on a double or triple chainring, and off you go.
The article Scott mentioned, by the way, appeared in Vintage Bicycle Quarterly, vol. 1, issue 3. It shows an Integral bike, with many design features very different from modern bikes. The floating chain probably is the most conventional part of the bike. (How about twin down and seat tubes, with the chainrings in between, or hubs where the bearings are located in the dropouts?)
Jan Heine Editor/Publisher Vintage Bicycle Quarterly http://www.mindspring.com/~heine/bikesite/bikesite/index.html