Star Ratchet and variations
This design in it's simplest form is used by DT Swiss. This design incorporates easily replaceable ratchet plates that offer the added benefit of every engagement point transferring torque. A more complicated variation is what is used in Chris King hubs.
Mechanism - DT Swiss

1. end piece
2. bearing
3. thread ring
4. axle
5. conical spring
6. star ratchet
7. rotor body
The DT Swiss system consists of two ratchet opposed ratchet plates that reside in the hub body and the freehub. These are held against each other by a pair of conical springs. When the rider pedals, the teeth engage as the freehub ratchet plate is turned via the splines along its circumference. This transfers power to the ratchet plate in the hub, and this in turn transfers power to the hub via the splines along its circumference.

There is a very basic animation here that shows the action of the ratchet plates.
Upgrading and servicing
One of the biggest advantages of this system is the ease with which you can upgrade, replace, or service the ratchet plates. This is actually a tool-free repair! The most basic DT Swiss hubs include an 18t ratchet plate, offering 20 deg engagement. This can be upgraded to a 36t (10deg) or even a 54t plate (6.67deg). This can be done to lower end DT wheels, so it is possible to take a less expensive wheel and gain a much higher engagement if desired. This also allows replacement of all contact surfaces (aside from the splines within the hub which are unlikely to wear or strip) to prolong the life of the hubs.

Advantages and Disadvantages
Advantages
- All contact points are transferring torque, always
- Easily serviced and wear components easily replaced
Disadvantages
- High contact leading to higher wear between contact surfaces
- The plates are pushed together by a spring, not rider torque, so they are more likely to skip as more torque is applied (contrast with Am Classic and Chris King), although I haven't personally ever heard of this happening.
- Relatively low engagement for cost on many wheels
Variations - Chris King
This is again one of the more complicated freehub mechanism designs. At it's core, it is a star ratchet system. There are a couple key differences:
- Both ratchets are inside the hub, allowing them to be larger
- The drive plate (the one timed to the freehub body) is the one furthest from it
- the drive plate is sprung, while the other is fixed inside the freehub body
- the 'ring drive' mechanism which forces the two plates together under torque
1.spring 2.drive plate 3.fixed plate
left: drive plate, right: fixed plate
What truly sets this apart is the ring drive mechanism. This is accomplished via helical grooves on the freehub body, which engage the drive plate. As torque is applied to the freehub, this forces the two plates together. As more torque is applied, the plates are forced together more strongly, effectively eliminating slipping or skipping.

drive ring shown on top with helical grooves
Many people would probably say that Chris King hubs are overkill - they are extremely well made and make use of many bearings that should contribute to their longevity (and your mechanic's headaches). At 72 points of engagement (5 deg) they can engage about as fast as any hub out there. Also, because the plates are forced together by rider torque, the spring can be weaker are cause less drag/wear and the teeth can be smaller without stripping (faster engagement). The cost/benefit can be hard to justify for many people, but if it makes you feel nice to have a well made hub on your bike, it's hard to do better than Chris King.