Ethereum is a blockchain with a computer embedded in it. It is the foundation for building apps and organizations in a decentralized, permissionless, censorship-resistant way.
In the Ethereum universe, there is a single, canonical computer (called the Ethereum Virtual Machine, or EVM) whose state everyone on the Ethereum network agrees on. Everyone who participates in the Ethereum network (every Ethereum node) keeps a copy of the state of this computer. Additionally, any participant can broadcast a request for this computer to perform arbitrary computation. Whenever such a request is broadcast, other participants on the network verify, validate, and carry out ("execute") the computation. This execution causes a state change in the EVM, which is committed and propagated throughout the entire network.
Requests for computation are called transaction requests; the record of all transactions and the EVM's present state gets stored on the blockchain, which in turn is stored and agreed upon by all nodes.
Ether (ETH) is the native cryptocurrency of Ethereum. The purpose of ETH is to allow for a market for computation. Such a market provides an economic incentive for participants to verify and execute transaction requests and provide computational resources to the network.
Any participant who broadcasts a transaction request must also offer some amount of ETH to the network as a bounty. The network will award this bounty to whoever eventually does the work of verifying the transaction, executing it, committing it to the blockchain, and broadcasting it to the network.
Blocks are batches of transactions with a hash of the previous block in the chain. This links blocks together (in a chain) because hashes are cryptographically derived from the block data. This prevents fraud, because one change in any block in history would invalidate all the following blocks as all subsequent hashes would change and everyone running the blockchain would notice.
A reusable snippet of code (a program) which a developer publishes into EVM state. Anyone can request that the smart contract code be executed by making a transaction request. Because developers can write arbitrary executable applications into the EVM (games, marketplaces, financial instruments, etc.) by publishing smart contracts, these are often also called dapps, or Decentralized Apps.