Skyrim:Invisibility (effect)

Invisibility makes the target become completely invisible for the duration of the spell.  Although you cannot be seen when you are invisible, you can still be heard, meaning enemies may still detect you.  You are more likely to be detected if you:
move too close to an enemy.
swing a weapon (even if you don't hit someone with it).
run into a mobile object or walk on a trap.
make noise, including casting the spell without using the Quiet Casting perk.

Invisibility will end if the caster:
Activates an item (including: using levers and buttons, opening doors and containers, grabbing and stealing items).
Uses a magical effect other than invisibility (including spells, potions, and scrolls).
Interacts with a character (including attack, pickpocket, and dialogue).

Alchemy
The following alchemy ingredients can be used to create a potion of Invisibility:
Ash Creep Cluster 
Bittergreen Petals  ( ×22px|Duration, ×Value)
Blister Pod Cap  ( ×22px|Duration, ×Value)
Chaurus Eggs
Crimson Nirnroot
Daedra Silk  ( ×22px|Duration, ×Value)
Elytra Ichor  ( ×22px|Duration, ×Value)
Flame Stalk  ( ×22px|Duration, ×Value)
Glassfish 
Green Butterfly Wing 
Ice Wraith Teeth
Luna Moth Wing
Nirnroot
Redwort Flower  ( ×22px|Duration, ×Value)
Screaming Maw  ( ×22px|Duration, ×Value)
Vampire Dust (1st effect)

Spells
Invisibility: Invisibility ( ;  ) is the only spell available in the base game that bestows this effect

Fade Other: Fade Other ( ;  ) also gives this effect but is not available without use of the Console

Fenrik's Welcome : ''Fenrik's Welcome'' ( ;  ) is a spell  that applies this effect, and can open locks of Expert-level or lower for 5 seconds.

Orum's Aquatic Escape : ''Orum's Aquatic Escape'' ( ;  ) is a spell  that applies this effect, along with a Water Breathing and Night Eye effect, for 15 seconds

Notes
Some possible exploits and innovative uses for the Invisibility effect include:
Stealing objects (not pickpocketing): even if the Invisibility breaks when you steal an item, you don't get any bounty for stealing.
Picking a lock won't break invisibility, but opening the locked door or chest will. You don't get a bounty for lockpicking, though.
Pickpocketing in broad daylight: Invisibility will break, but only after you've committed the act. It will not prevent you from getting caught if you fail, however. 
Helping sneak past opponents' lines of sight: with the help of Quiet Casting, you can use Invisibility to sneak right in front of opponents.
Breaking combat: if you manage to put enough distance between you and current opponents to not be able to be heard with the help of your Sneak skill, you can cast Invisibility to make them lose track of you and return to their former spot after a few seconds.
Returning to hidden mode while in plain sight of enemies if you have the Shadow Warrior perk.
Walking through a door or cave opening that loads into a new area while simultaneously charging up an Invisibility spell will complete the spell on the other side. This can be very useful for hiding yourself when first entering a new area that might have danger.
With the spell charged or a potion at the ready, one can perform any action then immediately (re)engage the Invisibility effect by drinking another potion or releasing charge on the spell. This applies to any object that can be activated, including doors, containers and levers.
Snap Attack: Cast and hold--don't resolve--Invisibility, attack an opponent, and immediately release Invisibility. You will visible for a split-second, then disappear again. Slaughter crowded rooms without breaking sneak.
Feeding as a Vampire doesn't break invisibility, so it can be useful for avoiding detection in a room full of potential witnesses.

Bugs
 
 
Saving and exiting the game or reloading from the main menu seems to fix this. Using the console on PC to readjust your race also seems to fix this.

See Also
Muffle
The Shadow Stone
Shadowcloak of Nocturnal
Embrace of Shadows