Annealing involves heating the neck of the cartridge case so that it is softened, making it more... maleable is the right word, I think... maybe ductile.
Shooting and resizing work hardens the neck, making more resistant to resizing and more likely to crack, either during sizing or firing.
Essentially annealing involves putting the case in a container of water up to about 5 mm below the bottom of the shoulder and using a blow torch to heat the neck and shoulder until it's dull red, then tipping the case over into the water to quench it.
On a lot of military cases you'll see multi colored streaking in the brass in the shoulder and neck, blues, purples, and reds. This is evidence of the factory annealing process.
This normally isn't seen in commercial cases because after they are annealed commercial cases are cleaned to remove those marks. Civilian shooters like nice, bright brass colored cases.
NEVER EVER heat the main body of the case, nor the head. These areas of the case have to be hard to resist the pressures generated when the cartridge is fired.
Here's an EXCELLENT article about the whys and hows of cartridge annealing:
http://www.6mmbr.com/annealing.html
While the author doesn't think much of the water pan method, it's perfectly suitable for someone who is shooting for fun, not for high precision accuracy, as the author of the article is.