Life> Death> Rebirth
In this world, everything that lives dies. However, those creatures with souls, such as human beings, have something more waiting for them. Souls are made of spiritual matter (also known as Reishi) which, like all other matter, can never be destroyed, only change in form. As such, there is a cycle of reincarnation, in which a human lives, dies, and is reborn again as an infant to start life once more. But there is a period in-between.
When a person dies, their soul leaves the body, and lingers upon the earth. Souls (and all Reishi for that matter) are invisible to the average living person, and as such, these lingering Souls are unnoticed by society, except by those rare few with spiritual powers, some of whom serve as legitimate mediums or religious leaders among a hosts of frauds. It is the Shinigamis' primary purpose to find these souls, and provide the 'soul burial,' using the Spirit King's power within to send the souls to their respective afterlives. For most, it's on to Soul Society, but for those few truly wicked souls, the flames of Hell await.
Regardless their Afterlife, a time will come when their tenure in the after-life is over, and the soul must await rebirth. A soul will only 'die' within it's afterlife of natural causes if it's time is up, and that person will be conceived into the living world immediately. When a soul is killed, however, it's Reishi gathers into a 'waiting realm' so to speak, a pocket dimension in which these souls wait for the next available conception without any consciousness or awareness. (After the great hollow hunt of the Quincies, which caused this dimension to grow near to bursting, the twelfth division created a mechanism to release certain pheremones into the living world to accelerate the rate of conception. It's this same mechanism that was responsible for the baby boomer generation that followed all the hollows killed after World War II.)
Not all infants born are recycled souls, however. When the Spirit King established the lifecycle, there was one facet for which he did not account (because at the time it didn't exist.) When a hollow devours a soul, the multi-verse can no longer recognize that soul's existence, and draws from the Spirit King's own power to form a new soul in it's place, which generates some of the comparatively rare births among couples in Soul Society (with the remaining portion being the deaths of the unborn.) Such a soul enters the 'waiting realm' as normal, and is conceived to the first couple to copulate in Soul Society after it's creation.