Soul

A soul (魂) is a record, the memory the world has; it is the second element. An existence independant from the body, it doesn't belong to the material world like the body, but to a concept above it, on an astral plane. It is the source of energy of a spiritual body.

Memories and Magic Circuits aren't stored in the body, but in the soul.

Nature
It is an existence that is eternal and indestructible in the material realm, but it cannot remain in this world without the body. Anchored inside the body, it can reshape itself within the body, but as it is fixated in the body, its fate is linked to the death of the body, and lasts only as long as the lifespan of the body.

When the body dies, the soul disperses and gradually fades away back to the Root to be recycled. This also applies to Servants, and magi need to feed them with prana to maintain them - something that shouldn't be doable normally on one's own, unless backed up by the Great Grail.

As it is a record, as long as the soul is undamaged, even if the body looses its genetic information, it is able to restore itself as it was. In Matou Zouken's case, his very soul was rotting: the body was restored as aged, and it started decaying again immediately.

Generally, the soul is only "what fixes the subject", a vessel. The soul is an element required to perform magecraft, but handling it is very difficult: to magi, it is an unchangeable, unreachable energy, therefore, meaningless to them.

The core that binds the soul is the head - the brain. It is said that there are three parts required for a lifeform: the soul, the brain, and the body.

The brain keeps the soul anchored to the body. The soul can reshape itself with the brain and the body. Usually, it is said that destroying either two out of the three elements is required to completely kill someone.

Roa and reincarnation
Michael Roa Valdamjong is probably the only one who ever succeeded in turning the unchangeable soul to a changeable information; it is how he acquired his nickname of Infinite Reincarnator (転生無限者). However, revival of the soul is difficult, and what really transmigrates is his tenacity more than the actual Roa.

As he can't actually transfer the whole 100% of his abilities, he reproduces them, and always ends up inferior to the previous incarnation.

Weight of a soul
Gilgamesh's soul weights the equivalent of hundreds of thousands times that of normal people. Also, according to Illyasviel von Einzbern, Matou Sakura could have been sustained as the Holy Grail if she had absorbed the souls of Saber, Lancer, Caster and Berserker along with Gilgamesh; this seems to imply that his soul weights as much as around three regular Servants, as the Great Grail normally needs the souls of the seven Servants to fully activate.

By the way, souls seem to have a color, and that of Gilgamesh is gold-colored. Gold also is a pretty heavy material, although that doesn't have to be related in any way.