On his quest to reach immortality, Lord Voldemort created horcruxes, but he didn’t know he accidentally created a seventh: Harry Potter and the Philosopher’s Stone, the first novel in what would become a series of seven books. The novels told the story of Harry Potter, known as “the boy who lived” for having survived an attack from the darkest wizard of all, Lord Voldemort, when he was only a baby.
Over the course of seven books (and later eight movies, with the final novel, Harry Potter and the Goblet of Fire. Readers and viewers later learned that Voldemort’s evil and ambition knew no limits, as he created six horcruxes so he could be, essentially, immortal. However, he wasn’t aware of the creation of a seventh horcrux, which ended up playing against him.
First off, a horcrux is an object that conceals a fragment of an individual’s soul, so if that person dies, a part of their soul will still be alive in that object. Anything can be a horcrux (including animals, as Voldemort turned Nagini into one), but inanimate objects work best as they’re harder to destroy – if a living being used as a horcrux dies, so does the fragment kept in it. In order to create a horcrux, the person has to commit murder (as this fragments the soul), then there’s a spell to separate the piece of soul and send it to the object/being, and then the final step is a mystery, but it’s said to be a horrific act that must be performed soon after the murder. Now, Voldemort went through all that six times – the diary, the ring, the locket, the cup, the diadem, and Nagini – and Harry’s case was very different.
Although Voldemort “killed” him, but he actually killed the piece of soul kept in him.
As for why Voldemort didn’t “feel” when that part of his soul was killed, many attribute it to desensitization: Voldemort was emotionally blocked on many levels due to being conceived under the effects of a love potion, and adding that he went through the process of creating horcruxes so many times, it came to a point where he couldn’t feel anything anymore, even if a part of his soul was being killed. Voldemort is the only wizard known to successfully create more than one horcrux, but it all came with a price, and ultimately, those horcruxes led up to his destruction.