Ethereum Classic (ETC) is a hard fork from Ethereum, initially developed as a response from a hack which took US$50 million in Ether from The DAO, an autonomous organization running a crowdfund to build open-source decentralized business models based on the Ethereum. An expressive group within the community aligned to pull off a fork which would roll back the chain to the state it was before the hacks happened, resulting in ETC.
IOHK, the group responsible for Cardano, is also one of the main ayers behind ETC.
Ethereum Classic and Ethereum, today, are similar platforms but with different futures ahead. ETC won't enjoy the same updates of Ethereum, such as the Casper protocol, but can as well be benefited by different enhancements by their new developers.