Dead stones are stones which don't have the slightest chance of escaping from enemy territory (sometimes, there's an exception; see comments below). It is assumed by both players that any more moves involving the stones will be futile, and they are removed at the end of the game.
In this invented example, the black stones are dead. If Black puts down a stone at the marked position, he is committing suicide for that area, and all White needs to do is put down a stone at the marked position to get rid of the black stones. Thus, the stones may be freely forgotten about.
See also: dead.
Alex Weldon: I wouldn't go so far as to say that dead stones can be forgotten about. In that example, they certainly can, but in general, dead stones often have aji. Sometimes they even come back from the dead.
Tamsin I'd like to proffer the following proverb of mine: "Stones are never truly dead until they're removed from the board".
As Alex pointed out above, even the deadest-seeming stones can still affect the progress of a game (aji). Only when they're completely captured does that effect cease.
Gringo: When using the Japanese rules, I'm (as White) tended to play at the marked spot. This would result in 2 points for the captured stones and 2 points for the area surrounded. Thus resulting in 4 points. Counting the dead Black stones as points as well as the marked spot results in 3 points. So why don't we play at the marked spot and take those 4 points?
Velobici: Compare these two cases in Japanese rules: White plays the marked point and has 4 points, two for the dead stones and two for the territory; White does not play the marked point and has 5 points, two for the dead stones and three for the territory.
Gringo: So that's how dead stones are counted! Thanks a lot.
axd: this page is marked as a beginner's page. However, experienced go players sometimes forget that beginners rarely understand why stones are dead - I'd even go further and say: even the case above is not trivial at all!
Trione: I was just teaching a friend to play and we ran across this situation. So, what's to stop Black here from disagreeing those two stones are dead, and thus forcing White to lose a point capturing them?
Patrick Traill: It rather depends what rules you are using, but some, such as the AGA rules, actually require a player who passes to give their opponent a so-called pass stone to add to their captives. Some people feel that area scoring is more natural, but that the territory counting method is easier; in that case it is fairly easy to see that pass stones let you use territory counting to the same result as area counting. (Scoring refers to the rules that define the result, and counting to the method used to calculate it.)