Souls in Slightly Damned are essentially the spiritual life force that every lifeform possesses and appear to be a key component to maintaining balance in the world.
The soul of a Median is tied to whatever realm they are in, be it the living world of Medius or any of the realms in the Afterlife. In Medius, the body and soul are as one and as such any damage the body may sustain will carry over to the soul at the time of death. For while in the Afterlife, they are not affected by the mortal limitations of self-nourishment, ailments or other conditions that would impede them. Instead, what may simulate the experience of death would just cause their soul to be restored the following in the Afterlife.
For a soul to reach the Afterlife, naturally the individual has to die and their soul would automatically go to where they are meant to according to the Great Book of Records. Sometimes though, for one reason or another, a soul will not automatically go to where they should be so Death would have to go and retrieve the soul to give them a proper judgement.
While it would appear that souls are created from the time a Median is born into the living world, they may also be created through the process of recycling souls together to create a brand new soul. The process seems to involve Death, who after allowing souls to spend enough time in either Heaven or Hell, he takes their souls and fuses them together. That soul will then be born into whatever form it happens to take, while presumably having small influences of their past selves to make up their new form.
Of course, because Death has not pass this ability to his substitute, the process of reincarnation cannot occur and as a result, the souls would remain in the Afterlife indefinitely. Based on what little information was given by Demons like Lakritz, it's possible that an overabundance of souls may end up destroying the Afterlife or make it uninhabitable if that's a cause for everyone to leave there.
Besides the fact that some souls get stuck in Medius without aide from Death or someone with his power, it seems that a soul that is brought up to Medius by way of an Ascension ritual, or brought down through its counterreaction will have their body and soul restored (in a way), where they seem to regain their original bodily functions like they never died. One side-effect to this is they seem to gain the ability to see Death or those with his power.