According to the IEEE Standard Computer Dictionary, System testing is testing conducted on a complete, integrated system to evaluate the system's compliance with its specified requirements.
System testing falls within the scope of Black box testing, and as such, should require no knowledge of the inner design of the code or logic (IEEE. IEEE Standard Computer Dictionary: A Compilation of IEEE Standard Computer Glossaries. New York, NY. 1990.).
Alpha testing and Beta testing are sub-categories of System testing.
As a rule, System testing takes, as its input, all of the "integrated" software components that have successfully passed Integration testing and also the software system itself integrated with any applicable hardware system(s). The purpose of Integration testing is to detect any inconsistencies between the software units that are integrated together called assemblages or between any of the assemblages and hardware. System testing is more of a limiting type of testing, where it seeks to detect both defects within the "inter-assemblages" and also the system as a whole.