Soulfire uses part of the human soul to enhance the user's magic. The soul regenerates over time and through acts which are "good for the soul", but will kill the user if it is completely drained.
Soulfire gives structure to magic. For example, it could theoretically allow a path to be made into deeper Faerie by stabilising the gateway. While angels use this, it is unknown if they use it on its own or bound to a form of magic.
Despite being similar to Hellfire in enhancing magic, there is a key difference: Hellfire gives power to the means; Soulfire gives power to the intention . If Dresden uses a spell meant to send a wave of force and knock someone down, Hellfire has made Dresden's wave of force stronger. On the other hand, Soulfire made a silvery hand that Dresden could control, with the same power of the original spell but given direction like a hand (rather than the simple 'push' that it gave before).
In the series Edit
In Small Favor, Soulfire was given to Harry Dresden by the archangel Uriel, after he had lost Hellfire when the Shadow of Lasciel (Lash) was erased from his subconscious mind at the end of White Night. Soulfire has manifested as a silvery ghost of Harry's force spell, and a blue-white supercharged version of his fire blast. Harry was told by Bob about its use of the human soul.
In Changes Harry used "holomancy" to create very realistic animated illusions of his companions.
In "Cold Days" Harry has an epiphany that soulfire is fueled from our entire being - yes from what we call our 'soul' but the amalgamation of what makes the Self - that we are a product of every decision made, every thought we have, every action done - its our entire being and identity that fuels Soulfire.