Most of these descriptions are taken from in game help.
To get new spells, clear dungeons and get astral chests.

Mass Heal	5 mana
Immediatly heals all allied heroes during battle. Heals 50 HP. (works correctly)
Power Up	5 mana
For the next 10 turns any hero who attacks will deal double normal damage.(Doubles DPS. 'next 10 turns' means your next 10 attacks. It's great against heavy armored enemies.) (works correctly)
Vampiric Power	6 mana
For the next 10 turns any hero who attacks will be healed for the amount of damage dealt('next 10 turns' means your next 10 attacks. Mass Heal or potion spam is better. works correctly)
Earth Barrier	6 mana
All allied heroes are gifted additional stackable physical protection for the rest of the battle(+3 physical def per a cast. works correctly)
Magical Shield	6 mana
All allied heroes are gifted additional stackable magical protection for the rest of the battle(+3 magical def per a cast. works correctly)
Cold Snap	6 mana
Immediatly strikes all enemies with Cold Energy granting a 50% chance to make them miss their next turn (NOT works. Enemies never miss their attacks)
Fire Rain	7 mana
Immediatly rains down high amounts of magical damage on all enemies. Can't be used to kill enemies. (Unleashes 60 DPS magical attack against all enemies. It's not effective to high magic-defense enemies.)
Tuition	7 mana
Heroes will receive double normal Experience for this battle (works correctly)