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How is Ethereum similar to Bitcoin?
I think it’s easiest to understand Ethereum by exploring the similarities and differences between Ethereum and a simpler system, Bitcoin. So what are the similarities?

Ethereum has a blockchain
Like Bitcoin, Ethereum has a blockchain, which contains blocks of data (transactions and smart contracts). The blocks are created or mined by some participants and distributed to other participants who validate them.

You can explore this blockchain here: https://etherscan.io

Blocks form a chain by referring to the hash or fingerprint of the previous block. See a gentle introduction to blockchain technology for a primer.

Ethereum is public and permissionless
Like Bitcoin, the main Ethereum network is a public, permissionless network – ie anyone can download or write some software to connect to the network and start creating transactions and smart contracts, validating them, and mining blocks without needing to log in or sign up with any other organisation.

In general, when people talk about Ethereum they mean the main public permissionless instance (version) of the network. However, like Bitcoin, you can take Ethereum software, modify it slightly and create private networks that aren’t connected to the main public network. The private tokens and smart contracts won’t be compatible with the public tokens though, for now. For more on the difference between public permissionless and private permissioned networks, see confused by blockchains? Revolution vs Evolution

Ethereum has Proof-of-Work (PoW) mining
Like Bitcoin, mining participants create valid blocks by spending electricity to find solutions to a mathematical puzzle. Ethereum’s PoW maths challenge called Ethash works slightly differently to Bitcoin’s, and this allows common hardware to be used for mining. This reduces the efficiency edge of task-specific hardware known as ASICs, which are common in Bitcoin mining.

For more on mining see a gentle introduction to bitcoin mining.

On Ethereum’s roadmap there is a plan to move from electricity-expensive Proof-of-Work mining to a more energy-efficient Proof-of-Stake protocol called Casper, in a future release of the Ethereum software called Serenity.

Ethereum has an inbuilt cryptocurrency
Ethereum’s token is called Ether, shortened to ETH. This is a cryptocurrency that can be traded for other cryptocurrencies or other sovereign currencies, just like BTC. Its current value is around US$13 per ETH token (Oct 2016). Token ownership is tracked on the Ethereum blockchain, just like BTC ownership is tracked on Bitcoin’s blockchain, though at a technical level they track them in slightly different ways.

For more on cryptocurrencies and tokens see a gentle introduction to digital tokens.