The blockchain is a distributed, public ledger that contains the history of every bitcoin transaction. Anyone can download a copy of the blockchain, and it can be inspected to trace the path of bitcoins from one bitcoin transaction to another. It should be noted that while there is a record of every bitcoin transaction ever made, these transactions are not inherently linked to real life identities. For this reason, Bitcoin is considered pseudonymous.
Bitcoins themselves are not files stored on your computer's hard drive like MP3s or PDFs. Rather "owning bitcoins", means owning a bitcoin address, which has a balance recorded on the blockchain. What it means to own a bitcoin address is to control the associated Private Key, and therefore allow the signing of transactions.
What is a Block?
A Block refers to a set of Bitcoin transactions from a certain time period. Blocks are "stacked" on top of each other in such a way that one block depends on the previous. In this manner, a chain of blocks is created, and thus we come to the term "blockchain".
Finding and publishing new blocks is what Bitcoin miners do to earn bitcoins. Whenever a new block is broadcast, approximately every 10 minutes, a quantity of bitcoins is received by the miner who solved that block. Bitcoin miners keep the network secure, and this is how they are rewarded. This system ensures that all transactions are valid, and keeps the bitcoin network secure from fraud.
If you've ever waited for a new bitcoin transaction to be confirmed, you were waiting for a new block to published containing your transaction. When that happens, the bitcoin network has deemed your transaction valid. Coinbase currently requires 3 network confirmations before the transaction is considered finalized, however this number will vary with other Bitcoin services.
How can I view the Blockchain?
There are many 'block explorer' services which allow you to look at what's in the blockchain. One example is https://blockchain.info/.
You can look at recent blocks here: https://blockchain.info/blocks
Or you can look at a specific transaction here: https://blockchain.info/tx/5384e1e61b5a6a800450267a163e64129e90bb557cda788186a6c9ad76f4cc9e