A computer system is the combination of hardware and software. A typical computer system has memory and set of states that define the relationship between the systems inputs and outputs.
Although in earlier times the personal computer or PC was an example of a computer system, in practice a person considered a Computer Systems Engineer these days deals with a range of different computer architectures.
Even the simplest computer classifies as a computer system, because at least two components (hardware and software) have to work together. But the real meaning of "computer system" comes with interconnection. Many computer systems can interconnect, that is, join to become a bigger system. In industry these days the interconnection of Variable speed drives and Programmable Logic Controllers (PLC's) to automate production lines and control motors is a common example of a computer system.
Interconnecting different computer systems can prove difficult due to incompatibilities. Sometimes these difficulties occur due to incompatabilities between differing hardware and sometimes between differing software. Technically knowledgeable people can configure disparate computers to communicate using a set of rules and constraints known as protocols. Protocols attempt to precisely define the communication within and between computing endpoints. If two computer systems define the same protocols, they may be capable of interconnecting and become a part of a larger system.