Ethereum is a post-Bitcoin blockchain. One of the major differences with Bitcoin is that Ethereum is designed to be Turing complete: the blockchain can be used to run any arbitrary code.
The ability to run arbitrary code was built to allow for the creation of smart contracts. The basic idea with a smart contract is that the contract's code (running on the blockchain) enforces all the rules of the contract itself. No need for courts or external parties to handle disputes: the code just does the right thing.
A company called Slock.it created a smart contract called the DAO (distributed autonomous organization). The DAO was a venture capital smart contract. Anyone could submit proposals, owners of the DAO would vote on them, and if successful, fund them, taking some ownership of the proposed venture in hope of a payoff.
The DAO got people excited, and $160 million worth of Ethereum was invested in it. However, the DAO had bugs, and someone exploited one of those bugs to move a large portion of the DAO's Ethereum into another contract that the DAO owners couldn't control. This crashed the ETH exchange rate.
The "hack" split the Ethereum community: some felt that nothing should be done, the DAO contract ran correctly, just not in a way that was anticipated. Others felt that the DAO funds should be recovered and returned, and suggested a soft-fork followed by a hard-fork to make this happen. Both the Ethereum Foundation and Slock.it supported the fork option.
Eventually the soft-fork was released. The soft-fork stayed on the same blockchain (hence the soft) as previous versions, it just blocked transactions related to the "hacked" contract so the former-DAO funds couldn't be moved. However, once again bugs were found, making folks running the soft-fork vulnerable to DOS attacks. I don't know if it ever reached consensus.
A short while later, the hard-fork version of Ethereum was released. As the name implies, this version splits the blockchain into two, one continuing the existing blockchain and another that moves the DAOs ETH into a new contract that's going to be distributed to the former DAO owners. The forked blockchain also put ~$5 million in the hands of a curator to distribute as they see fit.
The hard-fork went live today, and it looks like the majority of miners have switched to it, which means the hard-fork has become the new Ethereum blockchain.