Fourier is basically a file format converter for crystallographic data files. This view is based on the point of view that Fourier coefficients and density maps are merely different formats of the same information. A Fourier transform is required to convert the data from one representation to the other but the information is conserved in the process. Since all TNT programs which read crystallographic data can perform Fourier transforms themselves and will accept either a set of Fourier coefficients or a map interchangeably this use of Fourier is not usually needed.
The other task of Fourier is to perform various operations on crystallographic data sets. It is the program used by the correct and report_hkl scripts. A set of functions of Fourier not accessible from the shell scripts is its ability to add, subtract, multiply, or divide any pair of crystallographic data sets. For example, you can ask for a map which is determined from the sum of two sets of Fourier coefficients. In fact you can add a map and a set of coefficients and the program will Fourier transform one of them and then add them.