If you are asking about testing in Software Engineering (SE) then there are only three types of testing. WhiteBox testing, GrayBox testing and BlackBox testing. There is no brownbox testing in software engineering.
Whitebox testing - It refers to the testing a system with full knowledge and access to all source code and other architecture documents.
Blackbox testing - It refers to testing a system without knowledge of specification to the internal workings of the system, access to the source code, and knowledge of the architecture.
Graybox testing - It refers to a testing system by knowing limited information about the internals of the system