What does the blockchain represent? What does the blockchain represent? Is it the current balance of
    each account or is it the complete ledger?
The blockchain is the complete transactional history of Bitcoin. It collects every transaction performed in blocks which each are linked to their predecessor block.

By parsing the complete blockchain, you can discover all unspent balances. These are usually tied to addresses, which represent users much like email addresses do.

Also see these two related questions: 


What is the “blockchain”?
Can someone explain how the Bitcoin Blockchain works?