"You would be protected by an ancient magic of which he knows, which he despises, and which he has always, therefore, underestimated - to his cost. I am speaking, of course, of the fact that your mother died to save you. She gave you a lingering protection he never expected, a protection that flows in your veins to this day. I put my trust, therefore, in your mother's blood. I delivered you to her sister, her only remaining relative."
-- Dumbledore (OP37)
Ancient Magic is extremely old and mysterious, and a wand is not necessary for it to work. It is part of the “magical-ness” of the universe.
References from the canon
- During Gormlaith Gaunt's attack on Ilvermorny School, Isolt Sayre and her husband were awakened from a magical sleep by the power of Love when their twin daughters started to cry (Pm).
- Dumbledore's protections around Harry were connected through his mother's blood to his only living relatives, the Dursleys. (GF33,OP37)
- Lily's sacrifice of herself for Harry created a protection Voldemort could not break through. In fact, the sacrifice infused his skin with that love and Voldemort and Quirrell when under Voldemort's control couldn't even touch him. (PS17) Voldemort made a mistake by using some of Harry's blood to resurrect himself because it meant that while he could touch Harry without fear, he also couldn't kill him (GF32,DH35).
- Dragons have thick hides full of Ancient Magic, "which none but the most powerful spells can penetrate" (GF20).