The Undying is an end game raid achievement for killing all the Naxxramas bosses with no deaths in a single raid week (officially termed a raid lockout period). Completing this achievement is criteria for the meta achievement [Glory of the Raider (10 player)].
On a technical note, for this achievement you are considered "fighting a boss" when you are in combat and a boss is engaged. Even if the boss is dead, but you are still in combat with his adds, or his debuffs are still wearing off, this still impacts the achievement. So on Thaddius and Grobbulus in particular, be careful not to kill anyone (or kill yourself) while the debuffs wear off.
It is also highly recommended to use VOIP technology, Ventrilo ("vent"), Teamspeak ("ts"), or the in-game voice ("voice chat"). This can help your raid quickly adapt to a disconnect, a bad pull, or any other unexpected events and avoid a death.
Since many of the fights are not time constrained by enrages, you may find this achievement easier to do in parallel with The Dedicated Few, as there are fewer people to worry about and heal up (failure to safety dance, disconnecting on a fissure, getting behind the ice block, etc).
Player disconnects that do not result in a death will not change your status towards this achievement
You do not have to have a full raid to complete this achievement.
You do not have to be present for all boss kills to complete the achievement and earn the title.
You must be present when the final boss, [83+] Kel'Thuzad is killed.