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.