Clone eggs are eggs that can be purchased through the Black Market.
Clone eggs are used to clone a monster in your party. When you use a clone egg, you'll be able to select five potential monsters to clone. Once you select all five, a spinning wheel will initiate with only the five monsters you chose. Tap it to stop it. Whichever one of the five monsters it lands on you will get a second of.
Clone eggs can be used on any monster in your monsterpedia, whether it is in your possession or not.
If you want a higher chance of getting a particular dragon, put its first three forms in the clone egg. Put the hatchling, dragonling, and dragon. This drastically increases the chance you'll get at least one of them from the wheel.