A group is dead when its owner cannot, playing first with correct play, make it live with two eyes or in seki or make a ko for life.[1] The status of a group at any time is alive, unsettled, or dead. At any moment there may be several unsettled groups on the board.
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; 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.
The correct way to carry out the attrition method is part of the theory of nakade; see almost fill. Mistakes are possible.
See also the Eyes Collection.
The definition is rather weak and examples would be nice on this page. (See Dead Stones for a simple example.)
A better definition would be the other way around:
A group is dead if the opponent can capture it even against strongest resistance.
A group has two eyes if the opponent cannot capture it because filling its second-to-last liberty would be suicide.
A group lives in seki if the opponent cannot capture it because filling its second-to-last liberty would be self-atari, resulting in two eyes for the group.
--victim
66.23.192.42: I agree - it seems to me that, since two eyes and seki are not the only way a group can be alive, being unable to attain them is not the only way a group can be dead. A proper definition would be that no matter what moves the group's owner makes, the other player can capture the group (although possibly at the expense of their own groups - gets a bit fuzzy there).
[1]
A dead group may be brought to life during the game via ko threats.