This question can be slightly confusing as it is too simple and too complex at the same time! As global markets have become more and more intertwined with each other, financial systems in today's world are highly complex. There are different types of banks, and there is a large variety of roles that they play in the global economy.
There was a time when a bank did the simple job of keeping your money safe for you, and using it as part of their corpus. This money was then used to make simple investments and give loans to people, locally, and then interest would be used to pay customers interest on their deposits, and make some profit.
Today, personal banks are capable of doing much more. You can send money abroad using electronic wire transfers, receive money, make different types of investments and have a number of different types of accounts.
Banks today are huge multinational corporations that are linked to each other in some way, and this means that your finances are part of a global network of money. Just like with other services, big multinational banks can provide better deals for customers than small local banks, so customers are left with little choice but to opt for one of the big banks.