Ethereum is a blockchain that was designed to allow developers to use blockchain technologies cheaply, effectively, securely, and anywhere in the world. The native currency of Ethereum is properly called the Ether (ETH), but is often referred to as Ethereum as well.
The majority of cryptocurrencies that you come across will be based on Ethereum (BitCoin being the primary exception), but all sorts of blockchain applications use the Ethereum blockchain. Ethereum is sometimes referred to as a 'world computer', as any programmer can develop smart contracts on the Ethereum blockchain, to be run whenever anyone is willing to pay the gas fee.
The primary innovation that separates Ethereum from Bitcoin is that Ethereum is Turing complete; it can indeed act as a computer, and run programs known as decentralized applications (dApps). It also has its own programming languages, the best known of which is Solidity. These programs are uploaded to the blockchain, and are executed by miners, who are paid in Ether for proof of work.
Ethereum was originally designed with the idea that once set free on the blockchain, dApps should run as written without retroactive interference; this is often expressed using the catchphrase "code is law". Unfortunately, it is approximately impossible to pre-plan for every edge case, and in 2016 a major hack known as The DAO event cost users of The DAO $50 million in lost tokens. The community decided to fork the Ethereum chain, with Ethereum Classic maintaining with the original code, theft intact, and the fork that now holds the Ethereum name reset to a point before the theft.
While Ethereum is the new, exciting thing, it appears that it simply cannot scale up to meet demand, and other blockchains are being developed to compete with it. Potential competitors include QTUM, NEO, Hyperledger, IBM Blockchain Platform, Azure Blockchain Workbench, Cardano, and Corda. Additionally, Ethereum Classic (ETC), created by the aforementioned hard fork in 2016 is still going strong.