I would have thought that a single gear was the same as a fixed gear, but that does not seem the case. What am I missing?
Can a fixed gear bike have more then one gear?
A single speed bike has a single speed freewheel threaded onto the rear hub. The freewheel allows the rider to coast.
A fixed gear has a single cog threaded on the rear hub, along with a reverse threaded lockring, to prevent the cog from getting loose. The cog has no freewheeling action, so any time the wheel is moving, the pedals are moving too. This setup allows the rider to even ride backwards.
Generally speaking, fixed gear bikes have only a single gear. There are some setups however where instead of a single gear at the rear a special dingle with two cogs is placed in, and a pair of chainrings is up front, and the tooth difference between the chainrings is the same as the tooth difference in the cogs. Then, the rider can change gears for different riding, however it requires dismounting, loosening the rear wheel, and manually moving the chain from one set of cogs to the other.
There are also some 3 speeds or other internally geared hubs that have been owner modified to be multi-speed fixed hubs, but they are very, very rare.
Both have one gear, but a "fixed" requires you to constantly pedal as the back wheel turns, a "singlespeed" has a freewheel hub, ergo allowing you to freewheel.
In the UK (I believe) "fixed gear" refers to bicycles with only a single gear ratio, whether they have a freewheel or not. That might be where some confusion comes from.
In general, what are known as "fixed gear" bikes have track hubs or otherwise modified rear hubs (see bottom bracket lockring + Loctite) which result in what Paul describes: when the rear wheel is moving, the cog is spinning, and that means that (usually, see the Shimano Front Freewheel system) the pedals are also spinning.
It is possible to have a fixed gear bike which has more than one gear: you could have a flip-flop hub which had two different sized cogs, both "fixed" in that they do no spin independently of the wheel. I believe that there are also manufacturers who sell track hubs which can take more than one cog on the drive side, however both of these solutions require you to remove the wheel to change gears.
To my knowledge, there are no fixed gear bicycles which can change the gear ratio with a derailleur. This is because the back-pedal force could break a derailleur in this situation. That doesn't mean that someone hasn't hacked something together though, so I could be wrong here.