So, just recently I realized that most cosmere systems of magic have to do with shardic intent. I know that people have probably already thought of this, but it came to me while I was reading Warbreaker. On Endowment's shardworld, magic allows people to endow items with gifts, namely, the gift of life and breath. Awakeners give items life, of course, but even common people can give their breath to the Returned. And the Returned give their breath to heal.
On Scadrial, Ruin's system of magic requires the destruction of metal for power. Preservation's preserves power to be saved for later. Hemalurgy sort of preserves power by transferring it, but requires the ruin of a living being. So it's kind of a combination of the two.
On Sel, stuff is different, due to both shards being shattered. I won't bother.
I'm still trying to work out White Sand. I haven't read them yet.
On Roshar, a Knight Radiant's power is directly linked to being honorable by following the ideals of their order. Cultivation cultivates people with her magic. She nurtures with a gift, but prunes with a curse, in an attempt to foster healthy growth. Odium is... Odium. It seems like the Fused are fuled by passion, which is Odium's thing. We really don't have much concrete information on them though, except that they're basically uncool and Unhinged versions of the Knights Radiant.
That's... All I've read of the cosmere so far. Feel free to add, or correct.