Salvaging is the process of extracting a soul from a human, the core of an Evangelion, or Seed of Life and placing the soul into a new host (again, a human, Evangelion core, or Seed). In the event that the host body in question is missing, the body can be generated from LCL during the salvaging process. 
A salvage operation is not to be confused with a Contact Experiment; the two differ greatly. While the latter entails a human “diving” into the core of an Evangelion, setting their soul inside the core in the process, the former involves removing a soul from its current host and stabilizing it within another. More specifically, a Contact Experiment involves direct human contact – hence the name of the procedure – with an Evangelion’s core, while salvaging only involves the movement and placement of souls, and not necessarily into a core at that.
Soul extraction from a host presumably involves the generation of a weak Anti A.T. Field to free the soul, then the use of special salvage equipment to install the soul into its new host.[Source needed] Should the host have already lost cohesion (e.g. Shinji within Eva-01’s Entry Plug in Episode 20), no Anti A.T. Field generation would be necessary, as the soul's A.T. Field would already be neutralized. The appropriate equipment includes the following :
- A torus-shaped electromagnet that presumably functions in anchoring the soul’s position.
- A device that can emit radiation within the Evangelion’s core space.
- Sensors that keep track of the LCL temperature, the soul “forces” (i.e. Libido and Destrudo), the core pulse rate, blood type, and other relevant measurements.
While the above are at least implicitly described during the Episode 20 salvage attempt, the roles they play are never explained. Interestingly, it seems that the will of the soul can affect whether or not the salvage proceeds successfully; that is, the soul has to be willing to leave the present host for the new one. 
Instances of Salvage Operations
Nerv attempted to salvage Yui’s soul following her Contact Experiment with Eva-01, but this attempt failed. It is heavily implied that the birth of Rei 1 is somehow tied to this event, and Rei is known to have Lilith's soul and an appearance similar to Yui. Similarly, Nerv attempted to salvage Shinji’s soul after he fused with Eva-01 during the battle with Zeruel, but it too failed. Mysteriously, the core of Eva-01 spat out Shinji shortly after this failure. The exact mechanism behind this is never revealed but Shinji is seen floating towards Yui's soul inside the Eva, and it is possible that Yui herself decided to give "birth" to Shinji in his old body.
Nerv salvaged Rei 1’s soul after she was killed by Naoko Akagi. Her soul was presumably split, with one part placed within the core of Eva-00, while Rei 2 received another part.  After Rei 2 died during the battle against Armisael, her soul was salvaged and installed in Rei 3, seemingly along with the Rei 1 portion that belonged to Eva-00. 
Adam’s soul was salvaged and placed within Kaworu’s body prior to the events of Episode 24. It’s also safe to assume that, at some point between Kaworu’s death and the events of End of Evangelion, Adam’s soul was extracted from Kaworu’s remains and placed back into the Adam embryo, as Adam's(Kaworu's) soul is part of the giant Rei-Lilith being during Third Impact.
- See Maya’s statement in Episode 20: “Salvaging means reconstituting [Shinji’s] body and stabilizing his soul within it.”
- The Magi apparently control and monitor much of this, as they’re necessary to conduct the procedure. See the following Episode 20 dialogue:
Maya: “Salvaging means reconstituting [Shinji’s] body and stabilizing his soul within it.”
Misato: “Is that possible?”
Ristuko: “With the Magi’s support, yes.”
- See Ritsuko’s statement as the Episode 20 salvage failed: “Don’t you want to come back, Shinji?” On a related note, it seems that Yui’s salvage failed because she didn’t want to leave Eva-01.
- See the section The Second Angel under the Classified Information.
- See Ritsuko’s statement in Episode 23: “This is top secret. Scavenge the plug. Dispose of the parts concerned.”