For the next few posts I am going to geek out on sport climbing. If you're not a climber (or a sport climber), these are not for you. All of this information is in the context of sport climbing on trustworthy protection - not trad climbing!
How to be an awesome belayer
There are two reasons to be a good belayer - first, it takes one to know one. Second - you'll understand how safe (or unsafe) it can be for you when you are climbing, because you will learn what to expect when climbing.
A good belayer always has the right amount of rope out and when the climber falls. As a general rule, the "right amount of rope" means having a small droop of slack between you and the first bolt on the climb. Additionally, you want to be standing within one large step of the spot on the ground that is directly beneath the first bolt. Do not walk way off to the side. One large step. No farther.
Next, no matter what sort of device you're using, keep the belay device in the brake position. If you are using a Gri-Gri, use it the way it was intended. Do not hold it in your hand the whole time you're belaying! It's a hard habit to break (I held it in my hand for years... and finally stopped after getting a good demonstration from a Petzl rep. Just do it right.)
If you're using a tube-style belay device, keep it in the brake position at all times unless you are taking in slack. It is possible to feed slack quickly, without ever coming out of the brake position.
Be good at feeding and taking in slack. When you feed slack while your climber is clipping, step in towards the wall, to get an extra few inches out. Immediately return to the brake position. Do not short rope your climber. It makes him wonder if you're paying attention, or, if you are paying attention, it makes him wonder if you know what you're doing.
When your climber falls, give a soft catch. To give a soft catch, step in towards the wall and give a hop up the wall, just as their weight hits the rope. If you do it right, you will float a few feet up the wall. This makes all the difference for the comfort of your climber.
There will be situations where your climber will feel run out. He may be trying to clip having problems getting the rope through the draw, or maybe there is no good clipping position, or the clipping position is very high relative to the bolt. In all of these situations, your climber is aware of how far he will fall if he comes off the wall. Your job is to understand what to do if this happens so you can minimize the distance that he falls while still giving a very soft catch. The solution is simple - if your climber falls with a lot of rope out, while they're falling, pull in a full arm-length of slack, lock off the belay device, and then as their weight hits the rope, jump up the wall.
This is a big topic, however. Belaying, and learning to trust a good belayer, is a long topic. All of these steps can be taught, and refined. It takes practice, however, in an environment where it is safe to fail (but don't drop your climber, obviously) and where the odds are stacked in your favor to quickly learn and perfect each of these elements.
Next, we will talk about how to systematically build a very high degree of confidence in each of these areas.