An OS is probably something you associate only with computers, however even devices such as mobile phones run a kind of operating system.
An operating system basically manages the hardware while presenting you a comfortable way of interacting with a device. On computers you would have used Windows, which lets you manage file and folders with a mouse, and lets you manage and run software on your computer.
Similarly in mobiles there is a software which manages contacts, and SMSs, and making calls with the hardware on the phone. It is essentially a mobile operating system!
It is not even something that is only present in the latest and greatest of mobile phones. An operating system is something that is present on every mobile phone. Although newer mobiles have operating systems which allow for installing software just like you would on a computer.
Some of the popular mobile Phone operating systems are Symbian, which you can find on nearly every Nokia mobile phone, and many Sony and Samsung mobiles, the immensely popular iPhone runs a version of Apple's Mac OSX, called the iPhone OS. Palm's devices have used Palm OS, and their newer models run webOS. Even a version of Windows called Windows Mobile, and many of you may have heard of Android, Google's new operating systems for mobiles.
If you are purchasing a new Mobile Phone it is important to know which operating system it runs. Some of the cheaper phones are not capable of running third party applications, so if you need that functionality it is better to check beforehand.