Cole is the son of a male human (Benjamin Turner) and a female demon (Elizabeth Turner) making him half-human half-demon. But when Cole was born his mother killed his father and took her son from him.
In the third season of Charmed Belthazor was sent by the Triad (the most powerful demons in the Underworld) to kill the Charmed Ones. But Belthazors human side (Cole Turner) fell in love with Phoebe Halliwell and eventually betrayed the Triad and seemingly vanquished them. Through the rest of season three Cole used his demonic powers to fight demons alongside the Charmed Ones and became a very strong force of good.
After Cole vanquished the Triad he became an outlaw in the Underworld and was constantly hunted by demonic assassins. A major point of the fourth season was how close Cole and Phoebe became and how much their love grew, which was made clear when he proposed in episode eight 'Black as Cole' of season four. In 'Black as Cole' Cole also lost his demonic powers when the lover of one of Belthazor's victims used a potion which took away the demonic powers Cole had from Belthazor.
Cole started to lose himself unaware of who he really was when he couldn't fight in the battle of good vs. evil. In episode thirteen 'Charmed and Dangerous' The Source of all Evil unleashed the Hallow (an ancient magical force which almost obliterated the universe) which took the powers of the Charmed Ones. When Cole was approached by the Seer (a demonic psychic who has assisted many Sources) she told him that the only way the Charmed Ones would survive the attack from the Source after losing their powers to the Hallow was by allowing the Hallow to possess himself, which Cole did. With the powers of the Hallow Cole was able to protect the Charmed Ones long enough for them to be able to use a vanquishing spell to kill the Source. But when both Phoebe and the Seer combined forces at the end of the episode to trap the Hallow back in a magical box somehow Cole maintained the powers of the Source.
Once Cole had the powers of the Source he married Phoebe and eventually the human side of Cole was overtaken by the Source. But in the next few episodes Paige started to notice that Cole may be still be a demon, making her a target for Cole. Cole even almost executed a plan that would have killed Paige, but at the last moment pulled back showing that Cole still had some will over the Source. Cole also started to pull Phoebe away from her sisters and got her pregnant. Eventually once Phoebe realized that Cole was the Source at first she wanted to leave him, but then the Seer enticed her to accept Cole for their baby and Phoebe became the Queen of the Source. When Phoebe realized that she couldn't become a true force of evil she left Cole and she and her sisters vanquished him.
After the death of Cole the Seer came after Phoebe and Cole's baby, which was actually truly the child of the Source and would have been a great force of evil. But like Phoebe the Seer couldn't contain the power of the baby and the Seer along with the baby were vanquished when the Seer tried to tap into all of the baby's power to kill the Charmed Ones.
In the last episode of season four 'Witch Way Now' it was revealed that Cole's soul was actually still alive in the demonic wasteland because only Cole's demonic side 'Belthazor/the Source' died. Cole started to collect powers of dead demons in the wasteland and gained enough magic powers to resurrect himself. After resurrecting himself Cole also saved Phoebe's life from a witch hunter.
Now resurrected and one of the most powerful demonic forces in the world all Cole wanted was Phoebe back. Throughout the season Cole tried many different ways to get Phoebe back one even including mummifying Phoebe until he found a way to win her back. Obviously Cole started to become very desperate and a little insane.
In the twelfth episode 'Centennial Charmed' Cole became an Avatar (an ancient group of magical beings who want to bring the world to Utopia and ending the battle between good and evil) and changed the course of history by killing Paige before Piper and Phoebe discover she exists. But in this alternate universe Cole also doesn't have all of the new powers he gained in the wasteland making him immortal, and he is only Belthazor. Paige orbs into this alternate reality and joins forces with this other realities Piper and eventually Phoebe and vanquish Cole by using the Belthazor vanquishing potion. Because Cole had used his Avatar powers to transport himself to this time-line he was vanquished for good in the mainstream reality.
In the sixteenth episode of season seven 'The Seven Year Witch' Piper slips into a comma after being poisoned by a demon and makes a discovery. Piper finds that in this comatose form she is on another plane where Cole is cursed to. After giving Piper relationship advise about her and Leo she eventually survives do to this advice and is able to break out of her coma. In the end of the episode Phoebe's love interest of that season Drake (a former demon who gave up his demonic powers to spend one year as a human and then die) dies and for a brief moment is on the same plane as Cole. This final scene of the episode shows that Drake was actually holding up a bargain with Cole to seek out Phoebe and make sure she never gave up on love. In Drake's final mission he succeeded holding up his end of the bargain to Cole.
Powers And Abilities.
Cole had many powers over the years, changing each series.
As Belthazor, Cole had high resistance and could only be vanquished from a peice of Belthazor's own flesh. He could also Shimmer, throw Energy Balls, Conjure Athames and possessed Super Human Strength, as well as Spell Casting, Sensing, Adjusting to Piper's molecular immobilization, Telekinisis and Pyrokenisis.
Source of All Evil -
As the Source Of All Evil and ruler of the Underworld, Cole harnessed the gifts of Flaming, Immunity, Sensing, Fireballs, Regeneration, Force Fields, Electrokinesis, Pyrokenisis, Shapeshifting, Telekinisis, Summoning, Mind Control, Possession, Conjuring, Molecular Manipulation and Demonic Projection.
Back From the Wasteland -
When Cole returned from the Wasteland, he had thousands of mighty powers, but only showed 15 on screen. These were - Fading, Freezing Time, Energy Balls, Fireballs, Telekinesis, Pyrokinesis, Force Fields, Shapeshifting, Regeneration, Reconsitution, Conjuring, Mind Control, Summoning, Transformation, Demonic Projection, Molecular Manipulation, Chrono-Telekenisis, Sensing, Invincibility, Immortality and Power Granting.
As an Avatar -
When Cole became an Avatar, his abilities became unlimited, as the Avatars are the most powerful beings in the Universe. Here are the ones that were shown on screen by the Avatars - Reality Warping, Energy Waves, Spell Casting, Immortality, Invincibility, Lightning Bolts, Fire Throwing, Aerokinesis, Illusions, Chronokinesis, Summoning, Sensing, Removing People From Existence, Resurrection, Regeneration, Molecular Manipulation, Teleporation and Time Travel.