The Volunteer is the XCOM designation aptly given to the psionic soldier who volunteers to interact with the Ethereal Device and activates the Gollop Chamber. This character is essential to obtaining and participating in the final mission.
The Volunteer is a player chosen character with the stipulations that the soldier must be psionic and equipped with Psi Armor. Upon activating the Gollop Chamber, the Volunteer becomes unable to take off the Psi Armor and will receive a full three ranks of Psionic Powers (if not already obtained), plus the use of the Ethereal psi-ability Rift.
The Volunteer is referred to as "The New One" by the Uber Ethereal throughout the final mission.
The Volunteer must be taken as a squad member on the Temple Ship Assault and must survive the mission but otherwise remains fully under the player's control during the mission.
It was presumed by most (if not all) players that the Volunteer sacrificed him/herself to save the Earth by operating the Temple Ship via the Ethereal Device and detonating it in orbit. However, lead designer Jake Solomon revealed via Twitter that the Volunteer in fact did not die, but rather "disappeared" at the last second, having possibly teleported themselves off of the Temple Ship. It is yet unknown where the Volunteer is directly after disappearing from the Temple Ship, however it is confirmed that they survived.
If one examines the final cinematic carefully, the Volunteer can be seen disappearing in a flash right before the Temple Ship's destruction (though his/her shadow remains). However, due to the scene being extremely brief, many players were left with the impression that the Volunteer was killed; however, Jake Solomon confirmed that the cinematic was supposed to show that the Volunteer disappeared before the explosion.
If the Volunteer did indeed teleport off of the Temple Ship as is suggested, then this implies that the Volunteer is more gifted than any other psionic soldier, perhaps even exceeding the abilities of the Ethereals. Despite this, players do not have access to any teleporting ability using Psionics at any point in the game, leaving this assumption open to debate. (However, given that the Avatars of Xcom 2 possess this ability, confirms that this assumption is true)
However, due to XCOM 2 being set in a world where the Aliens successfully invaded Earth, The Volunteer does not exist in the canon. As the Aliens defeated XCOM around the time MECs' were researched, it is very unlikely that the Gollop Chamber was ever constructed, or even the Psionic Labs.
The Volunteer appears in XCOM: Enemy Unknown. Their name, gender, nationality, appearance, rank and class are completely up to the player's discretion. They will, however, always be Psionic and as such cannot be a MEC Trooper.
As Volunteering requires an individual to be Gifted, Annette Durand or Shaojie Zhang are often the prime candidates as they are almost always (always, in Annette's case) guaranteed to have Psionic capabilities.
The fact that the "volunteer" entering the Gollop Chamber is picked by the player implies that every available psionic soldier volunteered.