DPR - Delphi Project File. This is actually a Pascal source file; it just happens to be the main program for the application.
PAS - In Delphi, PAS files are always the source code to either a unit or a form. The main program of an application is in the DPR file.
DFM - These files are always paired with PAS files. The DFM file is the binary data used to set up initial data for components (IE, the properties you set in design mode rather than in code). You can't edit a DFM file with a text editor, but if you open it in Delphi, you will see a textual version of the contents.
DCU - A compiled unit, similar in concept to an OBJ file.
OPT - Project Options; i.e. compiler and linker settings, which form is the main form, what icon to use for the application, etc. Generally, the stuff you edit under Options/Project.
RES - A Windows resource file; generated automatically by Delphi and required by the compilation process. You don't need to worry about this file, but don't delete it either.
EXE - All of the above linked together into runnable format.
~DP - A backup file of the DPR file before the last save operation.
~PA - A backup of a .PAS file.
~DF - A backup of a .DFM file.