Despite what the name of the item might suggest, the Lucky Egg isn’t used like other eggs in the game are used. The Lucky egg is a consumable item that you can eat and it will give the player a buff that will cause your character go gain double XP for 30 minutes after consuming it. This can be incredibly powerful to use early on in the game when there are a lot of new Pokemon that you have not caught yet.
Lucky Eggs are not an item that you can find. In order to obtain one you must spend real money in the store to purchase the egg.