The owner of a rune is the player who has that rune in their runedock. Loosely, it's whoever had the rune to begin with, so in the case of champions it's who deployed the rune. If you are the owner of a rune, you are said to own it, which is required for some runes such Sacrifice (spell)
If you Possess a champion, the controller changes (you control the champion) but the owner does not change. This distinction prevents spells and effects like Repurpose and Forbidden Fruit from becoming too powerful.
For example, you can't possess an opposing Unstoppable champion, equip Forbidden Fruit, and then destroy another opposing champion, because both champions have the same owner (your opponent in this case).