There are 2 primary types of dyno's: a chassis dyno and an engine dyno. They both use a form of a "brake". They are generally not friction brakes like you would find on a car, but usually hydrodynamic brakes or eddy current brakes. Hydrodynamic brakes are basically pumps that have their output restricted by a servo valve of some sort. Eddy current brakes are essentially electric generators that have an electrical load placed on the output.

Chassis dyno's measure (to use the vernacular, for those of you with front wheel drive) "rear wheel horsepower". This is effectively what the car puts to ground. The car's drive wheels are placed on large rollers, which in turn drive the brake. It takes into account losses from the transmission and axle, rolling friction of the tires, etc. They are generally abusive to tires (don't bring you car to a chassis dyno unless you don't mind beating on the tires) and require fans, etc. to make sure you get really good airflow through the engine compartment. It's very easy to overheat on a chassis dyno, especially if you are doing a lot of tweaking.

Engine dyno's measure just the engine, not taking into account losses from the rest of the drivetrain. The engine is mounted on a cart or stand of some sort and the brake is coupled directly to the engine. I worked at a place that had an engine dyno (SuperFlow 7100 w/2000 ft/lb brake) to do performace testing for race cars. It was a hydrodynamic brake that used water as the pumped medium.

Running an engine on a dyno almost never (at least it's not supposed to!) destroy the engine. In fact most of the engines we tested/tuned went straight from the dyno to the race car. In some respects it's actually easier on the engine that smashing your foot down on a straightaway. The runs are generally over within 10 seconds, and there are no impact loads (such as shifting gears) on the engine. I ran my own engine on the dyno for 3 days straight, at least 10-15 dyno runs per day to work out exhaust, ignition, and fuel tuning. It went right from the dyno to my street car with no ill effects. And the best part was the engine was already broken in!

Cheers!