Blockchain refers to the technology of distributed digital ledger. It is a system where all computers connected to the system have information about all the transactions that have occurred on the network. Not only these computers store the information, they also verify each new transaction that occurs on the Blockchain. Any new transaction that is verified by each of the node (computer) is attached to the existing chain individually. Here a majority algorithm works and once a certain percentage of nodes approve the transaction the chain is updated. This way the Blockchain system ensures that it is hackproof and is available irrespective of any random computer going down. However, Bitcoin is just one of the implementation of Blockchain. The reason why it is used interchangeably with Blockchain is because it was the first Blockchain implementation. However, today we have many more Blockchain implementations like Ethereum and Ripple and therefore it is not proper to use the name Bitcion to refer to the Blockchain technology.
Bitcoin or Ether are currencies that are used in Bitcoin and Ethereum respectively. These currencies are mined by the nodes by solving complex caculations. This solving of calculation is referred to as Proof of Work. Once the proof is submitted, it is relayed to all nodes in the network. Other nodes drop their calculations of the same challenge and take up a new same challenge and this way mining goes on. When a new transaction has to happen, you need this digital currency to get your transaction processed. This is a way of paying the miner nodes for their work in validating your transaction. The amount of cryptocurrency pay is distributed among miners and the chain goes on.