They're a hub with a planetary gear system built inside (so automatic transmission style, rather than stick shift, if you know cars). The only exposed gear is a regular single speed cog. This gives you many of the advantages of single speed, like the sturdier chain and straight chainline, but you also get gears. There are several sorts of shifter variations, but most of the ones sold new use a twist shifter.

The Sturmey Archer 3 speed design is old enough that you can look up a lot of the servicing documentation online. Real nifty engineering. The fancier ones are newer, so there isn't as much in the way of public manuals. Sheldon Brown has a bunch of IGH information, and his gear calculator covers them.

If you poke around, there are a bunch of internal gear focused threads here too.