Daedra hearts are a rare drop from Daedra mobs: Atronachs and Dremoras.
They will not drop from conjured Daedra, though: so no farming your conjured servants.
Good places to find Daedra are during the Daedric quests The Black Star and Pieces of the Past .
In addition to that: One can be stolen from the Nightcaller Temple in the Daedric quest
Waking Nightmare One can be stolen from Kodiak's room in Jorrvaskr in Whiterun Enthir in Winterhold College
sells 2 every 48 hours at a huge premium
They will rarely show up for sale on random vendors throughout the world
The Wikia suggests a few other places as well : Alchemists in Orc Camps will sell 2-3 of them Babette at the Dark Brotherhood Sanctuary
One at Hall of the Vigilant directly south of Dawnstar and near Red Road Pass where a giant exists Random chance to be on any of several vendors in Markarth Side
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The only reliable way to guarantee that you'll get a Deadra Heart sadly involves giving up a Black Soul Gem.
First off, you'll need to do the Conjuration Ritual quest at the College of Winterhold - which means you need a Conjuration of 90.
You'll receive a Sigil Stone as a reward, which can be used to upgrade the Atronach Forge.
A Black Soul Gem and a Human Heart in the forge will yield a Daedra Heart.
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As an alternative to the vendors mentioned in other answers, Herluin Lothaire usually has two daedra hearts.
He restocks every 48 hours although is not guaranteed to restore both hearts after this time, instead replenishing with other rare ingredients such as void salts and fire salts.
The longest I have ever had to wait for at least one heart is 48 hours.
Herluin Lothaire's shop is located in The Ragged Flaggon (Thieves Guild Bar room) in Riften, however only after completing the second of the special jobs for Delvin.
Special jobs are obtained by doing 5 random jobs for Delvin or Vex in the same city.
Membership of the thieves guild is required, therefore, to use him as a vendor.
This means it may require more effort initially to set up this source of daedra hearts, though he forms a much more reliable one than drops from Daedras encountered in the wild.
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If you do Azuras quest and turn her soul gem back over to her, you travel inside the gem (crazy, I know) and you fight 4-5 daedra and get their hearts, once killed.
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After you complete the quest Pieces of the Past - going along with Mehrunes Dagon, He will spawn 2 Daedra in Daedric Armor, there will also be 2 more waiting for you inside the shrine (of Mehrunes Dagon) , and they will respawn everytime you visit - 4 Daedra =
4 Daedric Hearts - every time!
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I actually got a daedra heart dropped from a thalmor conjurer in the Ratways today.
My first one as well.
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You can buy Daedra hearts from Enthir in the College at Winterhold.
He will respawn 2 every 48 hours.
This is how I managed to make my armour without having to trawl around, waiting for them to be dropped.
I'd buy two, then wait or sleep for 2 days.
Then, rinse and repeat.the first time I did this, Enthir offered to sell me goods, no problem.
On a different playthrough though, he won't yet sell me anything.
Not sure what you have to do to make him sell you stuff, but I do know he becomes a fence for the Thieve's Guild.
Although, I was not a member of the guild the first time, so not sure if this affects whether he will sell to you or not.)
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I've learned quite a bit of smithing now and just started to be able to create Daedric armor and weapons. But now I noticed that I need Daedra hearts as an ingredients to create those.  Where can I find some Daedra to rip their hearts out and make some armor and weapons out of them?
Where can I find Daedra hearts?
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You can find them at the Black Star and Pieces of the Past. You can also steal one from the Nightcaller Temple as well as from vendord like Herluin Lothaire.  Daedric creatures, certain quests and specific vendors are the most prominent options.
Daedra hearts can drop from daedric creatures that are not conjured by the player. It is suggested that Daedra hearts could be found in several quests. Daedra hearts can be acquired from specific NPCs in certain locations or even by chance.