When the testing is done by the testing team, then the 
application has to be tested in thier real 
environment .That is called " system testing". For example 
bank has thier own tester (teller) to test the application.
There are so many types of system testing 
1 user acceptance testing.
2 end to end testing.
3 Functional testing .
4 load testing  etc.

Testing the software for required functionalities on 
intended hardware is known as system testing.

System testing is nothing but testing the whole application
which was integrated all the modules.
In system testing, there are two type of testing
1.Fuctionality testing
2.Non-functionality testing
Functionality testing means test whether application
functioning as per requirement or not.
In non-functionality testing, there are several types:
e.g Load, stress, performance, reliability, security,
usability,configuration, compatibility(forward & backward),
 scalability etc.