The way that Bitcoin actually works is like this:
Imagine a giant book that everyone has access to. Now this book has all of the records of who has how much money (Bitcoin). When you send any amount to anyone that information is recorded and broadcasted to everyone so that the records are all equally kept up.
Bitcoin is a public ledger system simply put. Within this ledger you have extremely secure information that only the holder of "private keys" can access. Each person with their own unique Bitcoin wallet (private wallet) has access to these keys.