I found myself at a Swiss crag with a guided climbing group. The group leader briefly demonstrated the use of a münter hitch for belaying and sent the first timers to fend for themselves. I watched as they torqued and pigtailed the ropes into a spaghetti'ed mess of münter madness. Still good in a pinch, this venerable knot has nothing on today's simple, bomber belay devices.
For the modern-climber's purposes, there are three major types of belay device: full-manual, self-braking, and mechanical-assist, each with its own inherent benefits and weaknesses.
Full-Manual
This category includes plates, tubes, or any belay device requiring direct belayer action to arrest a fall. They're versatile, lightweight, and afford-able, and create friction on a rope by bending it at two points. They're also good for trad climbing, when it's best not to shock-load the pro, as they give slightly during a catch. Available dual-friction designs can improve handling on thin or icy ropes. As "full-manual" implies, unhand the rope for an instant, and your partner could deck.
Self-Braking
These belay devices are similar to full-manual options, but can also be configured to allow a belayer to bring up the second(s) directly off the anchor point in "self-braking" mode (i.e., the belayer can catch a fall without a fully manual lockoff; still a bit of a misnomer, as belayer attention and control are required at all times with all belay devices). These are pricier than full-manual devices, but well worth it for multi-pitch rock or ice.
Mechanical-Assist
Mechanical-assists use intrinsic moving parts to help grab the rope when weighted. These devices can catch a fall even if the belayer is momentarily distracted or incapacitated. (Great for hangdog sessions, they also hold weight with little or no effort.) Mechanical-assist devices have become de rigueur for many gym, sport, and even trad climbers, but they do have some weaknesses: none currently allow for double-rope rapping or belaying, they're heavier than tubes or plates, and they cost more. However, they're well worth it if they suit your needs.
