Skyrim:Spell Absorption

Spell Absorption is a rare effect that simultaneously provides defense, by negating hostile spells, and restoration, by refilling your Magicka reserves. There are only three ways to obtain this effect (four with the Dragonborn expansion). Each of these uses a different variant of the Spell Absorption effect, but the effects stack with each other.
The Breton Dragonskin racial greater power provides 50% Spell Absorption for 60 seconds, once per day (effect form ID  )
The Atronach Stone provides permanent 50% Spell Absorption, at the cost of slower Magicka regeneration (effect form ID  ).
The Atronach perk from the Alteration skill provides up to 30% permanent Spell Absorption (effect form ID  ).
Miraak's Robes  provide 15% Spell Absorption, which can be strengthened up to 25% if also wearing Miraak's Gloves and/or Miraak's Boots.

Unlike Resist Magic which always reduces magic damage taken by a percentage displayed, Magic Absorption has a chance - equal to the percentage displayed - of completely blocking all damage. If it activates you take no damage from an enemy spell and, if the spell had a Magicka cost, you gain Magicka equal to the base casting cost of the spell. If it doesn't activate, you take full damage from the spell as normal. However, do note that Magic Absorption is calculated before Magic Resistance, so even if it doesn't activate you're still eligible for damage reduction from your Magic and Elemental Resistance.

Interestingly, Magic Absorption works not only with spells but also with every effect in the game that is not the following:
Normal damage from melee or ranged attacks
Falling damage
Drowning damage
Therefore it is possible for Magic Absorption to negate effects of shouts, poison, diseases, weapon enchantments, various debuffs, etc. However if the effect had no associated Magicka cost you will not gain any Magicka from it.

Getting 100% Spell Absorption
In the vanilla game, and without using glitches or bugs, a value of 100% is only temporarily possible. This can be achieved with the Breton racial power Dragonskin along with the Atronach Stone as the active Standing Stone. Other characters can have at most 80% Magic Absorption, using the Atronach Stone and Atronach perk from Alteration skill. However, there are two glitches to boost this to 100%:
 
 

The Dragonborn expansion adds Miraak's Robes, which can be combined with either Miraak's Boots or Miraak's Gloves, the Atronach Stone power, and the Atronach Alteration perk to get 100% absorption on any character.

Alchemy
The following alchemy ingredients can be used to create a potion of Spell Absorption:
Blind Watcher's Eye 
Comberry 
Daedra Venin 
Fire Petal 
Lichor 
Watcher's Eye 
Withering Moon 

Notes
The Destruction school is only associated with the Atronach variant of Spell Absorption (even though the Alteration school provides the perk). The other two variants do not have any associated school.
Spell Absorption grants the user the ability to absorb the magicka from the Circle of Protection spell granting you practically infinite magicka.  
The Ward Absorb perk from the Restoration school works similarly to Spell Absorption, but does not stack with it, as spells that hit the ward will not hit the user.

Bugs
 
When your own summon spell is absorbed, you initially lose the amount of magicka you spent casting the spell, the same as if you had successfully cast the spell.  However, no summoning portal or summoned creature appears.  At that point, you regain magicka equivalent to the base cost of the spell (as with all absorbed spells).  If your Conjuration skill level is high, you can regain significantly more magicka than you originally cast.  For example, it might cost you 32 magicka to cast Conjure Familiar but you will absorb 107 magicka.  Therefore, this bug can be exploited to refill your magicka (as long as you have enough magicka to cast a summon spell).
Reanimate spells will still work with 100% Spell Absorption, and can be used as a successful alternative to summoning.