Cartridge cases, in this case. In comments on the post about reforming .284 to 7.5 Swiss Chris mentioned this. I should have covered it in the same post, but got in a hurry and didn't want to put off posting it.
As in steel, annealing brass is to reduce the hardness of it. If you look at most military rifle ammo with brass cases, you'll see that the case neck, usually going down into the shoulders, is discolored; that's from annealing. Commercial ammo they do this and then polish the cases so they're nice & shiny all over, but most military contracts(so I'm told) specify that the annealing colors be there to prove the cases have so been treated, so they're polished, then annealed, then cleaned- not polished- and loaded.
When you reform a case you work-harden the brass, sometimes a lot. This leaves it stiff enough that when it flexes under the pressure of firing it can crack, so after forming or, as Chris notes, after the first firing it should be annealed.
I searched through various gun boards and found a lot of opinions and methods. This is the one I used on these cases. In my case, I formed the cases, cleaned off the case lube and then annealed them.
You need a propane torch, preferably one with a narrow flame, a pan or bowl for water and something to hold the cases with. This is done in a dim room as you have to be able to see it when the brass starts showing red.
Take the cleaned cases(burning case lube makes nasty smoke) and start the torch, bowl of water standing close. I sanded a piece of steel wire to fit into the flash hole of the primer pocket to hold the case. Stick it in the hole and hold the case neck in the flame, just ahead of the blue center, with the case angled so the flame is pointing toward the case mouth, not the body. Ideally the point of the flame should be hitting at or just ahead of the angle where neck joins shoulder. Turn the case constantly as you have to heat it evenly all the way around. As soon as you see the case neck showing red, drop it in the water. With steel the quench would harden it, with brass it keeps it from hardening a bit as it cools.
I'd suggest that as soon as the first few cases are cool enough, put them under a good light and look them over closely. You want to see the color difference caused by the process running into the shoulders, down to where shoulders meet body is fine, but not way down into the case body, and it should be even all around as to how far down it goes. When you've got it right, run the rest of the cases through.
That's it. This should relieve enough of the stress set up by the forming to leave the brass with enough spring to hold the bullet securely but not so stiff that it'll crack.
I've read some people saying to hold the case at the base, and even if you don't see color to quench as soon as it gets too hot to hold. I like something to hold the hot case with.
Whatever else you may do, DO NOT anneal way down into the body, or at the base and the 'web' above it. Those areas need to be stiffer, so leave them alone. If the annealing color runs down that far, you're better off trashing it.
There are a couple of companies that make machines specifically do anneal cases. If you were going to do a lot of them on a fairly regular basis one of these would be the way to go.