A small operating system is one that is made of a comparatively small amount of underlying code. As a result, it usually requires fewer resources from the machine it runs on.
As a very generalized rule, most systems described as a small operating system are based on the Linux kernel. The kernel is effectively the brain of a computer and decides exactly what it should do at any particular moment. Linux is normally used in open source systems, which vary widely and some are smaller than others. Microsoft's Windows and Apple's OSX are both usually considered very large operating systems.
Because the idea of a small operating system is comparative rather than absolute, there are editions of Windows that could be described as small. The best known of these is Windows CE, which is designed for very low-specification computers and for standalone devices. With the latter, the system is normally built directly into the device itself. Some devices that run Windows CE include interactive information screens and the seat-back entertainment systems on some airlines.
The main advantage of a small operating system is that it can work on a low-specification computer and usually does not require as fast a computer or as much memory as larger systems. This leaves more resources for running applications. It also makes it possible to use older computers for longer before they are considered obsolete.
Another benefit is that a small operating system, by definition, has less potential to contain bugs. This only applies in absolute terms, however. A badly written small operating system will likely cause as many or more problems than a well written large one.
Some operating systems are so small that they can be stored on, and even run from, a portable device. This could include a USB memory stick. This means that the operating system can be run instantly on a computer without the need to install it. This can be particularly useful when dealing with a computer where the installed system has been corrupted.
The drawbacks of a small operating system depend on why it is small. For example, it may only run on the command line, meaning information is inputted and displayed purely in text form rather than graphically, which can be offputting to many users. A small system may also have saved space by having more limited support for hardware devices such as printers or modems.