Ethereum is a decentralized platform that runs smart contracts: applications that run exactly as programmed without any possibility of fraud or third party interference.
These apps run on a custom built blockchain, an enormously powerful shared global infrastructure that can move value around and represent the ownership of property. This enables developers to create markets, store registries of debts or promises, move funds in accordance with instructions given long in the past (like a will or a futures contract) and many other things that have not been invented yet, all without a middle man or counterparty risk.
The project was bootstrapped via an ether presale in August 2014 by fans all around the world. It is developed by the Ethereum Foundation, a Swiss non-profit, with contributions from great minds across the globe.
Ethereum is an open source, public, blockchain-based distributed computing platform and operating system featuring smart contract (scripting) functionality. It supports a modified version of Nakamoto consensus via transaction-based state transitions. Ether is a cryptocurrency generated by the Ethereum platform and used to compensate mining nodes for computations performed. Each Ethereum account has an ether balance and act as gas to power transactions and smart contracts.
 Gas is used to pay for computation within the EVM during transaction execution. Accounts also act as storage for blockchain data (state) and have key-value pairs called storage.
The account model used in Ethereum has many differences from bitcoin’s UTXO model. In UTXO-based systems like bitcoin, one sends an output of a previous transaction to a new public key associated with a new recipient. By contrast, in account-based systems like Ethereum, one instead sends value from one account’s nonce to another account’s public key. This allows for simpler smart contracts where state transitions do not depend on UTXO inclusion.
 However, this also means that accounts can be re-used for multiple transactions which may confuse users not familiar with UTXO-based systems like bitcoin. For this reason, some people have called for Ethereum to switch its consensus algorithm back to Proof of Work so that it can use UTXOs instead of accounts; however, this would require a hard fork which may be undesirable given the current level of network stability and hashrate.
In short, ETHE is not an Ethereum but rather it is a cryptocurrency generated by the Ethereum platform.