I see a lot of people asking questions about this, so I figured I'd do a little guide to help out. Nothing crazy, just the basics you need to know to get your rare, unbreedable dragons.
You find these dragons in the BUILD menu of Dragon City under the EGGS tab. You will come back to this menu in Dragon City to claim the dragons after you complete the leveling process in SW or SE.
Part 1: Social Wars
Reaching level 10 in Social Wars unlocks the Robot Dragon. Reaching level 20 in Social Wars unlocks the Fossil Dragon.
Robot Dragon - LVL 10 in Social Wars
Fossil Dragon - LVL 20 in Social Wars
Getting to level 10 and 20 in Social Wars is very easy. Just do missions and build as many things as you can. I got both dragons in one sitting each, although getting from 10 to 20 is considerably longer.
Part 2: Social Empires
Reaching level 10 in Social Empires unlocks the Paladin Dragon. Reaching level 20 in Social Wars unlocks the Butterfly Dragon.
Paladin Dragon - LVL 10 in Social Empires
Butterfly Dragon - LVL 20 in Social Empires
Getting to level 10 in SE is pretty fast and easy. I did it in one sitting. Just build all you can afford to and do quests. Getting to level 20 in SE was much more difficult than SW and took me several days of playing to finish.
BONUS: Claim Powerful Mechs and Dragons to Make Leveling Easier
Remember that being high level in Dragon City gives you very powerful robots and dragons to use in SE and SW. Make sure you watch the side of your screen and claim them. It makes leveling up much easier.
OMFG, I HIT LVL 10/20 AND DRAGON CITY SAYS I HAVEN'T, YET! WTF DO I DO?!
Calm down, it's going to be okay! Just exit DC, open SW/SE and build a house or something. Open DC again and see if it worked. You might have to open and close the game a few times for the message to be sent to DC to unlock you dragons.