An operating system is the program that controls all the other parts of a computer system - both the hardware and the software. Most importantly, it allows you to make use of the facilities provided by the system.
Every computer has an operating system.
Strictly speaking, UNIX is not one single operating system, it is a family of operating systems. Although these are mostly similar, there are small differences which can cause problems. The most obvious examples are the layout of the file system and the exact format of certain commands. As its name implies, the kernel is at the core of each UNIX system and is loaded in whenever the system is started up - referred to as a boot of the system. It manages the entire resources of the system, presenting them to you and every other user as a coherent system.