How did we transfer money thousands of years ago, when we lived in small villages, knew and trusted each other? We bartered easily, as we do with cash today. But when money is transferred online, it can get a little tricky. Banks and credit card services solve this problem with a 'ledger system' ('cash system') which stores account histories and shows who owns what.
When John makes money transfers online from one person to another, it is the bank that makes the transfer from John to another person. John can't do it himself because there is a risk of cheating - he can copy and paste digital money (because it's just numbered on a computer) and send it to two different people. So, we trust the bank to send the money and make sure only one person receives the money. These banks can cheat, but we believe they are not cheating.
If the transfer occurs at the same bank, it's easy to do. However, if the banks are different it will be a little tricky as they may have different cash systems and need to be matched. Therefore, they charge a small fee, and these transfers usually take a little time. When these banks and financial systems come from different countries, things get more complicated: different languages, systems, currencies, more people have to coordinate, and so on, so transfer fees and times increase. That's why the current financial system is so complex. The system is a big mess of various cash systems that exist in the world.
Bitcoin changed all that. With what? Bitcoin is a cash system that synchronizes across the internet, so everyone can access the same cash account in real-time regardless of who and where they are. The result? Money can be transferred from one party to another without delays and exorbitant fees. Like when money was used before the world got so big and complicated.
What is the Bitcoin payment system used for? The answer: whatever you spend regular money on sending it to friends and family - both local and overseas, buy things online, receive a salary, and so on.