A soul, a creation of God, is an eternal spiritual essence of a person or being that persists and remains after its host, the body, dies. The soul is the source of a being's thoughts, emotions, sentience, and consciousness. Eve, the Mother of All, created monsters whose souls function similarly, but are destined instead to another dimension. In the Natural Order, human souls are transported to the afterlife by Death's servants, Reapers. The reaper Tessa refuses to tell Dean, on more than one occasion, where she takes the souls. Known destinations of souls are Heaven, Hell and Purgatory. The souls of angels who have been stripped of their grace undergo the same process of life, death, and afterlife as human souls.
The soul separate from its body is visualized as a blinding white ball of light. Both Death and Famine own a black case that can house human souls, securing them until they can manage the contained souls.
Souls are said to be indestructible and immortal. They can be damaged but can't be broken. Death claimed that not even he could break a soul. Although the fate of ghosts was at first unknown, it's revealed that the souls of the ghosts are taken to their final destination by Reapers after their remains are burned rather than going into oblivion as is commonly believed.
Crossroad Demons were the first to show the use of souls. In exchange for a soul, they would grant humans any desire they wish. Once the exchange was completed, the crossroad demons give the person ten years before they come to collect. It can be as little as one year (Dean's deal) to as large as twenty-five years (Becky's possible deal). Either way the demon has to collect and send the soul to Hell. Once in Hell, the soul is usually tortured until it has become so twisted and damaged, it is no longer human but a demon. Due to the fact that the demon starts out as a twisted soul, burning its original human bones will kill a demon like it would put a ghost to rest.
Death and Fairies know whether someone has a soul by just looking at the individual. However, Angels must first perform a painful procedure where the angel reaches into the human's stomach (Bobby called it a "soulonoscopy"), indicating that souls are tangible. Death being able to physically handle souls also substantiates this premise. Archangels like Michael and Lucifer can abuse them in their true form without using a Vessel.
Both angels and demons can use souls to gain more power. Castiel used Bobby Singer's soul to heal himself, he also used the power of 50,000 souls to blast Raphael, effectively starting the civil war in Heaven. When Castiel opens a portal to Purgatory using a ritual, he absorbed the souls of every monster there and the power he gained from them was enough to easily overwhelm and kill Raphael. The souls from Purgatory also made Castiel immune to the deadly effects of an Angelic Blade. With his own power drastically augmented, Castiel claims the title of God.
Apparently, Angels can absorb a soul's experiences/memories, sucking them into their allotted vessel, as Cas did for Sam.
Effects on humans
For humans, naturally, the soul gives them feelings and a conscience. When Castiel rescues Sam from Lucifer's Cage his soul remains trapped in the cage until Death retrieves it for him. Without his soul, he continues to have physical needs for food, drink and sex, he doesn't sleep, has a high tolerance for pain, and is completely emotionless. At one point, Dean angrily refers to him as being a robot.
Souls can also affect a person mentally. Because of Sam's soul being in the cage for so long, Castiel tells Dean it could make Sam a drooling mess if he gets his soul back. In order for Sam to survive his soul's return, Death erects "a wall" in Sam's head to help him survive. When Castiel destroys this wall, Sam is plagued with memories of his time in Hell and is nearly driven to insanity by a hallucination of Lucifer. Supposedly the spiritual scarring also leaves mutational damage upon the body; the Leviathan Sam, who recovered some of Sam's DNA also complains of the psychological issue, suggesting that the soul's damage can transfer into the DNA as well, as opposed to being confined to Sam's soul and consciousness.