Basically a zener is used as basic voltage regulators, just like the circuit above. They can also be used as overvoltage protection, in that a (short) pulse above their rated voltage will be absorbed instead of being passed to the device down the line in parallel with them. Often enough, that mode is used for FET gates, by sticking a 9.1V zener on there if they have a 10V gate max, so that spikes from noise, commutation, etc., don't end up blowing the gate and latching the FET on or something.
Exactly how they work inside I don't exactly know, though I'm guessing that the Wikipedia entry for them either explains that or it has a source link to a place that does.
Power ratings vary. I have some 5W zeners in various voltages, and they're around 3mm in diameter and maybe 5 or 6mm long. I've also got a bunch of tiny glass ones that are half a grain of rice, which are probably 1/8W if that, mostly pulls out of PC power supplies and UPS boards. There are sizes between those, and also even bigger ones.
I think the largest I have in anything is in a very old linear power supply from a teletype, and it is about 1/2" across, 3/4" long, and one end is a nut with the other end a threaded cylinder, basically being a bolt into a heatsink, with a regular nut securing it on the threaded end, one wire out each end. I dont' know it's wattage, but the power supply itself does 5V and 24V at something like 20 or 30A. The zener on each rail also has a resistor to it that is very large, with a metal heatsink that bolts to the big backplane heatsink. I'd guess the zener is 10 or 20W or more.