For an introductory notion of "dead", see dead - introductory.
The Chinese rules define life and death this way. "At the end of the game, stones which both players agree could inevitably be captured are dead. Stones that cannot be captured are alive."
The Ing rules give this definition. "Stones live or die according to whether they can be removed. Stones that can be removed are dead; stones that cannot be removed are alive."
The Japanese rules determine life and death at the end of play by hypothetical play with a special ko rule.
Note: Plain Area Scoring rulesets, such as Tromp-Taylor rules, do not have and do not need definitions of life and death. An implicit consequence of such rulesets is this: All stones on the board at the end of play are alive.
If your opponent's group is dead, you have succeeded in killing it, but not yet in capturing it and taking it off the board. That you may have to do, when one of your own groups is attacked; however, for the most part you don't want to spend plays capturing dead stones if you have a choice.
The method of capturing a dead group may involve plays inside the group, requiring the method of playing inside systematically, also known as attrition method because several plays inside may be required to remove one net liberty.
See also the Eyes Collection.
Black's stones are dead. Only two plays in a row at or a wrong answer by white may bring them back to life. (See Bulky Five)
There is some discussion about the pros and cons of various definitions at /discussion.
A dead group may be brought to life during the game via ko threats. I.e. taking the diagram above, black plays on one as a ko threat and if white ignores that, black may live by playing on the second .