Hatching
Some eggs can be hatched from the inventory. Luck usually does not apply to these, but Hatch Speed does.
Rifts and the Secret Bounty
Some eggs appear on temporary Rifts with a 5x to 25x Luck boost, and a few eggs, such as the 
Aura Egg and the Underworld Egg, can only be found on rifts.
One egg each day also carries the Secret Bounty.
All eggs
Every egg has its own page under Eggs.