The blockchain is the core of Bitcoin. It’s what makes everything works. It’s what makes a fundamentally different currency than normal money like USD or EUR.
The blockchain is the source of truth. The blockchain is the public ledger where all transactions ever made is stored and this ledger is stored on thousands of computers all around the world, which means that everyone knows exactly what coins are sent where and what the balance of all wallets are - but there is no way to know who owns those wallets.
Because the source of truth is spread across lots of computers, it is very difficult - almost impossible - to cheat the system and send some money to dont really have. Or claim that you sent some coins somewhere, if you didn’t send the coins. The blockchain knows exactly how much coins was sent and where they were sent.
Every wallet has a copy of the ledger. The wallets uses the ledger to calculate the balance on the wallet, by looking at all transaction to the wallet minus all the transactions sent by the wallet. The result is the amount of unspent coins in the wallet, hence the wallet balance. Desktop wallets has their own blockchain ledger and participate in the network. Online wallets like Blockchain.info or Coinbase also keep a copy of the ledger and since the ledger contains all transaction - to all wallets - ever made, they can use the same ledger to calculate the balance of all wallets on the system.
When you want to send coins from you wallet to someone else, you simply broadcast the transaction to the network. Before you can broadcast the transaction, your own wallet will check if the have enough coins on your wallet to make the transaction. When the transaction is broadcasted to the network, the other computers in the network will also check against their copy of the blockchain ledger, that you have enough coins to make the transaction. If the transaction is accepted by the network, it will become part of the blockchain, which means everyone in the network will have a record of that transaction, making it impossible to spend the coins again. And the is how the blockchain controls the flow of coins in the network and makes fraud impossible - or at least very difficult.
Also watch this great video, which does a good job of explaning how everything works.
The Bitcoin blockchain is currently more than 60gb. What if you want to check the balance of a wallet or see if a transaction was sent to a certain wallet? Blockchain explorers is the answer. Blockchain explorers lets you search the entire ledger and find transactions. You can find out when an transaction was made, how many confirmations it has and you can even see if the coins are still on the wallet or if they have been sent somewhere else.
These are some of the most popular Bitcoin blockchain explorers: