In days of old, people used to trade and barter things. People would swap a pig for a goat, a bag of wheat for some chickens, a 'magic' stick for all their worldly possessions (for the people of old were very superstitious). This would ultimately lead to massive arguments over the relative values of each of these things against the others.
The basic idea of banking, and the typical services offered, can be traced back to medieval times. The Knights Templar were known to store valuables, arrange loans, and manage the transfer of various funds between people and even countries. Modern banking originated in Europe in the 17th Century.
The bank is a rather curious concept. People give their money to an organisation which invests the funds wisely, making money for itself, but it returns only a fraction of the proceeds to the customer in the form of interest payments. However, banks do have their advantages. They allow us to keep our money securely, they can lend us money, they can purchase and exchange foreign currency for us and they can make one feel amazingly inferior, if that's your sort of thing.
The first of those points is actually a rather good one. Instead of hiding money under your bed, in a drawer or in your pants, you can sleep safe in the knowledge that your money is safe and sound. Even if a bank is robbed, you'll still be able to get your money back, and the cases of entire banking corporations going bankrupt are incredibly rare, at least in the West.
The remaining points, however, are less advantageous, and are usually a means for the bank to make money from you in a rather sneaky manner. The most obvious of these is when buying or selling foreign currency. Not only will the bank offer an exchange rate that is advantageous to them, they will also charge a 'commission' rate. Note that it's often not possible to exchange less than the equivalent of five pounds because you'd actually have to pay the bank for the privilege.
One recent phenomenon that has made people feel warmer feelings towards the banks is that they allow customers to get at their money through a 'hole in the wall'. Note that these aren't really actual holes in actual walls, but machines that dispense money after one inserts a plastic card and presses a few buttons. However, a bank's philosophy is this; if you should be so privileged as to receive one of these beautiful cashcards then you should take care of it much more than, say, a very expensive digital watch.