DNA Soul

A DNA Soul is a Reploid's operational DNA Program that has been pulled out by Erasure. The term was first coined by Iris, because as she explained, a Reploid's DNA is like a human's soul. As such the DNA Soul is synonymous to a Reploid's soul.

History
In Mega Man Xtreme 2, Berkana one of the Soul Erasers took souls from erased Reploids and inserted them for use in the bodies of Mavericks in order to revive them. She also gathered these souls in attempts to increase her own power and that of her partner Gareth by creating items from them.

When the Maverick Hunters investigated Laguz Island, they quickly encountered Gareth, who told them he was only erasing the souls of evil Mavericks. Knowing of the Erasure of innocent Reploids, however, X and Zero didn't believe him. They also found DNA data draining out of their enemies and quickly concluded what the Soul Erasers were truly up to. X and Zero decided they couldn't leave the souls of Mavericks alone, and also vowed to return the souls of the innocent Reploids that were erased. Assisting them on their mission, Iris used the DNA Souls obtained from the Mavericks, to create power-up items of her own. In the end, X and Zero succeeded in defeating the Soul Erasers, and successfully returned the souls of the erased Reploids. In the ending, the DNA souls were represented by floating glowing orbs, similar to Cyber Elves.

Though not Reploids, the Nightmare that appear in Mega Man X6 have a special type of DNA Soul, referred to by Alia as the "Nightmare Soul." After the Nightmare is destroyed in combat, the Nightmare Soul is what remains, represented by a blue floating orb. If this orb is not quickly retrieved, the Nightmare will resurrect itself. By retrieving these Nightmare Souls, X and Zero not only counter the Nightmare's resurrection, but they also raise in Hunter Rank, allowing themselves to equip more part. Furthermore, if enough Nightmare Souls are retrieved early on, Alia will succeed in her analysis of the Nightmare, allowing X and Zero access to Gate's Lab. X and Zero weren't the only ones collecting Nightmare Souls, however. Dynamo too became interested in the Nightmare Souls. He wanted to use the Nightmare Souls to power up as they're very nice DNA for that purpose.

Trivia

 * The introduction story in the manual for Mega Man Xtreme 2 contains a description of what DNA Souls are. However, this description is not present in the original manual for the game and contradicts the information presented in both the game, and other resources. According to this introduction story a new type of DNA chip was discovered on one of the Maverick Reploids found on Laguz Island, it is this chip that is called "DNA Soul". It's further elaborated that the chip contained DNA data duplicated from Maverick Reploids who were defeated in the past. Investigators suspected that it was used to revive them. As this new chip extracts DNA data from Reploids and reproduces it. Furthermore, it's also said to contain a program that produces a Sigma virus and then erases itself.
 * In the Mega Man Zero series, the term "Cyber Elves" is often applied to the souls of Reploids that continue to exist without their bodies, such is the case with X and Phantom from Mega Man Zero 3. As a matter of fact, in the early concept of the ZERO-series, Cyber Elves were called "Replisoul" and Ciel was considered a "Soul Summoner". TELOS further elaborates on the connection between DNA souls and Cyber Elves by describing Omega's "personality program" as a Cyber Elf.

I'd just like to add my tit-bit here.

What the actual FUCK capcom? DNA Soul? You mean to tell me that ROBOTS with DNA (Exclusive to living things) have SOULS in the DNA?