Elemental Contracts are one of the game's resources. It is very rare.
There is a 40% chance of being awarded an Elemental Contract after every 5 waves on Blitz Acres but also a 10% chance of obtaining them through Monster's Den boss battles from Big Behe onwards. You can also get them in events.
They are used to research magic skills and talents on the Magic Ward such as Hail to Blizzard and Ice Tempest, Static to Lightning Bolt and Lightning Strike, Militia to Royal Guard and Immortal. Also used for upgrading Magic Talents.
They are also used for high Level Soldier Upgrades.