Black Box verification refers to the technique of verification if system with no knowledge of the internals of the DUT. Black Box testbench do not have access to the source code of DUT, and are oblivious of the DUT architecture. A Black Box testbench, typically, interacts with a system through a user interface by providing inputs and examining outputs, without knowing where and how the inputs were operated upon. In Black Box verification, the target DUT is exercised over a range of inputs, and the outputs are observed for correctness. How those outputs are generated or what is inside the box doesn't matter.
White Box Verification
In White box verification, testbench has access to internal structure of DUT. This makes the testbench environment reuse less. This is not much preferred in the industry.
Gray Box Verification
Gray box verification, the name itself Conway that testbench has access to some part of the DUT.