Most of the damaging spells in The Elder Scrolls V: Skyrim comes from the Destruction spells which commands the forces of Ice, Fire and lightning.
Each element has it's bonus effects which can be used to counter a specific enemy.
Perks from the Destruction tree helps you improve the damage and effects of your choosen element.
Lightning Destruction spells damage target's health and half of that to its magicka, making it a good choice when fighting other mages.
Fire Destruction Spells do most base damage and also burns the target, dealing damage over time and amplifies other fire spells.
Ice Destruction Spells do the least amount of base damage but damages stamina as well as having a chance to paralyze the Target once upgraded.
The way you learn new Destruction spells depends on how high your Destruction level is, At the start of the game every race can buy Novice spells. Afterwards you'll be able to buy better spells every 25 level in the Destruction tree (Apprentice, Adept, Expert, Master). However to learn the Master spells ( Firestorm, Blizzard, Lighting Storm) you'll have to complete a quest from Faralda at The College of Winterhold, once complete your rewarded with Firestorm with the other two for sale in her shop.