This quest will be available during the Lilac and Gooseberries main quest, and can be started by visiting the Notice Board in White Orchard, then talking to Odolan, skipping the Notice Board and talking to Odolan directly, or by stumbling upon Noonwraith in Hovel.
Locate Odolan's cottage on the western outskirts of White Orchard and ask Odolan about the contract. When you're done, head south along the woodland path so you can find the spirit that haunts the well. You'll also need to find evidence of the spirit's presence using your Witcher Senses, which involves the following.
The Witcher 3: The Beast of White Orchard
The Witcher 3: White Orchard - Missing in Action
The Witcher 3: Velen - Bloody Baron, Family Matters
The Witcher 3: Ciri's Room, Fake Papers
The Witcher 3: Novigrad - Broken Flowers, Get Junior
The Witcher 3: Skellige - The King is Dead, Nameless
The Witcher 3: The Battle of Kaer Morhen
The Witcher 3: Bald Mountain, Imlerith Battle
The Witcher 3: Velen Contracts - Jenny 'O the Woods, Missing Brother
You'll end up with three new objectives to complete here, including find out what binds the Noonwraith to the ruined village, follow the drag marks, read the diary from the ruined house, and examine the well. Once again, all of these can be skipped by spotting the skeleton on the well wall and retrieving the bracelet. To do things the proper way, use your Witcher Senses to find the following items in the nearby area.
If you'd rather do things the long and proper way, go down the well and look for something that belonged to the dead woman. The dead woman being the one you saw when you looked down the well in the first place. Use your Witcher Senses to find the bracelet, then find a way out of the cave, and leave the cave by heading to the surface and swimming past a small waterfall. Climb up on the bank to the north and investigate the bracelet. When you're done, head to the east, through an underground tunnel that takes you to a pond. Climb out and make your way back to the village.
Prepare to fight the Noonwraith and light a fire near the well. To do this, consider reading up on it in the Bestiary, noting that it is vulnerable to a few specific things.
When you're satisfied that you're ready for the battle, go back to the well and light a fire to destroy the Noonwraith's bones and bracelet. This will summon the beast. Defeat the Noonwraith using everything you learned from the Bestiary, then inspect the corpse to take a trophy from the Noonwraith. The last thing you need to do to complete this contract is return to Odolan for a reward. Of course, that will depend on what you negotiated before started the contract.