in the start of the game when you open it you will se something that says mystery eggs on it it will say your friends name and level and then from lvl 0 to 10 it will say small eggs from lvl 10 to 20 it will say medium egg from lvl 20 to lvl 30 it will say big egg and 30+ super egg
It's free to send as far as I know. My only quibble is you can't unselect people from the list, and I have FB friends who HATE being bombarded with requests. Also, sometimes it pops up just as I'm harvesting gold or food and any click anywhere on the page will send it.
Once you get to a certain level (not sure where that line is) you start to get the Super mystery eggs that contain 10k , 600 and better dragons that Pivato mentioned. I don't think I got Super Mystery Eggs until I was around L-26 or so. It doesn't cost anything to send them. I keep 2 of each egg and sell the rest.