It's called thaumaturgy, Kvothe calls it sympathy. Basically, what he did was create a magical link between the hair and the person. Magic flows through one object and is attached to the other. As above, so below; you can make something happen on a smaller scale and if you feed enough energy into it, it'll happen on a larger one.
The best way to create these links are using blood, hair or nail clippings. Like a voodoo doll, which is what Kvothe made.
Thaumaturgic rituals can be used to kill people, if there's enough energy and intent behind the spell. If a person has your hair, they can do whatever they want to you. Assuming they have a strong enough mojo to funnel into it.