MATLAB Engine is a library to call MATLAB® software from standalone programs, for example written in C/C++ or Fortran.
While mex files are DLLs encapsulating C/C++ or Fortran code to be used by a main program written in MATLAB, sometimes the inverse is necessary: A main program in C/C++ or Fortran needs to access MATLAB's capabilities. This can be done using MATLAB Engine, which is documented in the MATLAB online help.
Note that a second way to use MATLAB from C/C++ or Fortran is to turn the MATLAB code it be used into a library using MATLAB Compiler.
Questions should have this tag if they are about the use of MATLAB Engine specifically. If the issue also occurs with standalone MATLAB the tag should be avoided and matlab be used instead.