As you have seen, computers have gotten very complex. One computer might have hundreds of programs stored on the hard drive. Each program might be able to read hundreds of documents. A program might also have all kinds of fancy devices attached to it like scanners, printers, and sound cards. How is this stuff organized? Is there some kind of central control?
Here are some more specific problems....
As you can see, the central problems that operating systems are there to solve are problems of organization. Mainly, operating systems help you to organize all the stuff on your disk drives. Operating systems are also invaluable in giving your computer an interface, or 'look and feel' that you can interact with efficiently. Also, operating systems help all the parts of your system communicate with each other.