Unless you're a big macher with plenty of cash to burn, to do any kind of serious competitive shooting you eventually realize that reloading is both a practical and an economic necessity. Part of that process is to re-use the spent brass cases. I'm currently shooting .308 in competition, and for a variety of reasons I prefer Lapua brand cases. I try to buy it on sale, but even then, it's roughly $0.65-$0.75 a case. A piece of brass can only be fired so many times before it become unusable, either because it cracks, will no longer hold a primer, or some other defect develops. So obviously, the more firings I can get out of each case the better.
Every time a section of the brass case is "worked" (being bent, expanding, shrinking, etc.), that portion of the brass will harden. Eventually it will harden to the point of becoming brittle and susceptible to cracking. When a brass case is fired in a firearm, it will expand to fill the chamber, and the case mouth in particular will expand to release the bullet. In bottleneck rifle brass, the shoulder will also expand and move forward. On many semi-automatic guns the effect will be greater than a bolt action, and of course the case can get dented during extraction or even stepped on once it hits the ground. Before a case can be reused, it has to be run through a sizing die that essentially bends and squeezes it back into shape, which works the brass, especially the case mouth, once again. Typically cracks will first appear in the case mouth, but also in the shoulder, because these areas get "worked" the most.
Annealing, as I understand it, is a process of heating the brass which causes the brass alloys to soften, undoing some of the accumulated work hardening. There are several benefits to annealing the brass, one of those being significantly longer life. How much longer depends on many factors, such as the specific alloy the manufacturer used, the rifle chamber, the dies, etc. But the anecdotal evidence I've read suggests that perhaps double or triple. While a typical Lapua .308 case might go 8-12 firings before showing a crack, annealing could double or triple that life, all else equal.
The second benefit of annealing is it results in a more uniform neck tension when reloading. As brass work hardens, it's "springiness" becomes variable, such that after a few firings to pieces run through the same sizing die could show considerable variance in neck tension (amount of force gripping the bullet). Annealing makes the neck tension more uniform, which (supposedly) can have a significant impact on consistent muzzle velocity of the reloaded ammunition, and thus accuracy. This is not a critical factor for me during the 200 and 300 yard stages, but at the 600 yard line consistent ammo is important. That said, I've yet to see the data substantiating the role that neck tension can play in consistency, but a lot of competitive shooters in High Power claim it is important to use either relatively fresh or annealed brass at 600 for best results.
So how do you anneal cases? Well first and foremost, let me say that like reloading, it can be dangerous if you don't know how to do it correctly. But that said, it is not rocket science, and anyone who has learned to safely reload can anneal. I'm not going to get into specifics here. Only that you never ever want to anneal the head of the case (the bottom of the case where the primer is) because you do not want to soften that area, which could lead to a catastrophic case failure and a big kaboom.
The general technique most people use is to expose the brass, typically just the shoulder and above, to a propane torch. There are expensive ($200+) turntable machines on the market, or on the other hand, simple techniques involving bowls of water and lazy suzans. You can search Youtube and find all sorts of techniques and devices reloaders have come up with. I decided to go with a kit called the AnnealRite:
Should everyone anneal? Probably not. First of all, some cases, especially small pistol cases, are probably too short to safely anneal. Secondly, many shooters feel that some of the less expensive brass, like 9mm, .45ACP and .223 can just be shot until it cracks, and then scrapped, typically after 4-12 firings depending on a number of factors. I know most High Power shooters feel that way about .223 brass, although I know a handful do anneal their 600 yard .223 brass on belief that more consistent neck tension will help with accuracy. Others simply use new .223 cases for the 600 yard line once or twice, then relegate that brass to shorter line work, replacing older brass that needs to be scrapped. Also, depending on the power of the load and make of brass, in some cases the primer pocket will lose its ability to hold a primer before the neck or mouth of the case hardens, rendering the annealing a largely wasted effort.
For a while I thought I could get away with only neck sizing the cases, which means simply tightening the mouth of the case to hold a new bullet. Reloaders find that this puts very little stress on the brass and it lasts a long time. However, after about three or four firings, the bolt started to get hard to close on my rifle--not so hard that it would be a problem with single-load stages, but in rapid fire it could cause problems. I also noticed that even with neck sizing, the feel of bullet seating became very inconsistent case-to-case with brass fired several times, suggesting that neck tension was inconsistent. So now I've started full-length sizing the brass more frequently, which will cause more work hardening. I know in my case Lapua brass has a good reputation for primer pocket durability, and I run mild loads. Lots of people report very good case life with annealing Lapua brass, as well as improved long range accuracy. So I decided to go for it. Propane torches are very cheap, as is Tempilac to get the flame exposure timing down. I'm running around 200 cases for two monthly 88 matches, so each case is getting fired roughly once a month. If I can double or triple the case life, that's $75 or so a year in brass cost savings, not to mention possible accuracy benefits. It is a very fast process, so it does not add too much to reloading time.
As a side note, even if you are not reloading your cases, you are a schmuck if you aren't saving them. A significant amount (if not most) of the cost of your ammo comes from the brass case. If you are leaving them at the range, the range is selling them off as scrap (or to reloaders) and you are essentially giving them money. Save your cases in a bucket, and sooner or later you will a) reload them yourself, b) sell or trade them.