Most climbers start out learning to belay off of their harness. For most snow climbs and most all instances of belaying a leader, this is still usually the best method. But for belaying a second when rock climbing, belaying directly from the anchor with either a Munter hitch or some version of an autolocking belay device has a host of advantages. Here’s a few of them.
Advantages . . .
better on difficult pitches (where fall is likely for the second), as it’s usually easier to catch and hold a fall
better for easy terrain (where second is moving fast and will likely not fall) as you can take in rope faster
puts less force on the anchor (only the weight of the second)
autolocks with Reverso, ATC Guide, or other modern belay devices (or just use a Munter hitch to keep it simple!)
belayer is free to move around more
easier to escape the belay and initiate a rescue
easy to rig a mechanical advantage haul to pull up the second if needed
easier to properly equalize the anchor toward the direction of load
Disadvantages . . .
fall directly impacts anchor (rarely a problem on rock if the anchor is stout.)
So, when might you want to belay the second off of your harness? Basically, when the anchor is anything less than 200% solid.
That means most of the time when you are snow climbing, and in many alpine rock climbing situations. When climbing alpine rock (which typically means many pitches over a long day and trying to move as efficiently as possible over relatively easy climbing) an "anchor" might be one decent cam plugged into a crack, or a sling around a small tree or over a rock horn, or some other single point of gear.
In this case, the belayer will typically sit down, try to brace their feet in a solid position, and belay off of their harness. By doing this, the belayer takes the impact force for any fall onto them, and the anchor is essentially backing up their stance.
Note: if you choose to use an auto locking blade device such as an ATC Guide or Petzl Reverso, keep in mind that there have been many serious accidents when people use these devices in correctly, often when trying to lower someone when the rope is under load. Be absolutely sure you know how to use these devices correctly before ever trying it on a real climb.