Souls are the essence of what makes a person. Every living thing has a soul.
Relius Clover views people for their souls alone, and can see the strength and weaknesses of each one. Some souls are strong enough to survive journeys through the Boundary, and can "point" in certain directions, indicating their strength. Weak souls are said to be vague and scattered, not having a purpose or goal. The strongest of souls attract the Azure, and combined with those who seek the Azure, both of these kinds of souls can gain Drives which are physical reflections of them. Being the Drive of The Origin, Izanami had debated on what a soul actually is, and if she possesses one - Relius confirmed that she does have one as she possesses the emotion of desire. 
For the Detonator line of weapons, a strong soul is required. Ada Clover and Ignis Clover are said to have incredibly strong souls, but they pale in comparison to the soul of Makoto Nanaya which points in multiple directions and has multiple layers.  Relius notes that a soul can be toppled and reshaped if its fulcrum is destroyed - in Makoto's case, this would be the deaths of Noel Vermillion and Tsubaki Yayoi.
Souls are shaped by experience, but not necessarily memories. The soul of Hakumen is said to be towering, but nothing is mentioned on the soul of Jin Kisaragi. This is also true of Nu-13 and Lambda-11, who have parts of The Origin's soul, but due to their experiences in life, the souls have grown and become independent.  Information about a person can be gleamed from looking at a soul if one possesses the ability, such as one's gender. 
In order to function, Nox Nyctores need multiple souls to form their core, and exceptional souls with control over life itself are needed to fuel the Kushinada's Lynchpin.   Noel's soul is a reflection of The Origin's, and also contains the souls of Izanami, Saya, and Mu-12 - the last of which she severed from her and became an independent entity for a short duration. Other individuals are capable of moving their souls between vessels; Tenjo Amanohokosaka possessed this ability and was able to move her body into the Phoenix: Rettenjo. 
According to Relius, the soul is an inorganic mass and the true form of a person. 
- BlazBlue: Central Fiction, Arcade Mode, Izanami, Act III
- BlazBlue: Continuum Shift Extend, Story Mode, Makoto Nanaya - Slight hope, Bad Ending
- BlazBlue: Calamity Trigger, Story Mode, Arakune - solve equation
- BlazBlue: Central Fiction, Story Mode, Episode 011, Chapter 88
- BlazBlue: Remix Heart, Chapter 19
- BlazBlue: Continuum Shift Extend, Story Mode, that which is inherited
- BlazBlue: Phase Shift 0, Chapter 05 - Sealed Green
- BlazBlue: Spiral Shift - Hero of the Frozen Blade, Chapter 05
- BlazBlue: Continuum Shift Extend, Story Mode, Relius Clover - Darkness Visible, True Ending