Agent Helena heals a unit 100 hp, and that unit retreats if able. If it can't retreat, it heals 100. She's a rare unit (3 star).
Prayer for Victory heals all units 20. It's uncommon (2 star).
Father Gwydion heals all units 25 with his special. He's a rare.
There's another rare Avalon unit (Paladin something) who can heal 20, but then he becomes the prime target next round. He's a herald.
Royal Alchemist can heal another unit for 25, but it isn't as spammable as Gwydion's. Additionally, any surviving Alchemists might restore themselves to full hp after a fight. They're commons (1 star).
Loyal Defenders can regain 25 hp with their special, but I find it to be close to useless since they only have 80. They're commons.
Song of Silver is the only other heal spell, and that heals the worst damaged unit for 100. I'm not too sure on the specifics as to what qualifies as "most damaged," though. It's rare.
There are a few drain effects, notably on Blood Acolyte, Death Dealer, and Void Scarab. Mistress of War has the only aoe drain I know of.
There is also a pseudo-healing in Veteran; Veteran units gain hp equal to their Veteran score after each combat. Usually, this is irrelevant, as they tend to take more damage than their Veteran score, but with enough distractions/units that get attacked first, you might be able to restore a decent amount.
Beyond those, all damage dealt is permanent. I may have missed one or two units, but I think that's all of them.