You must interface with the Windows Shell API module to let Windows know that your application accepts dropped files (this can be done in your main form's create event), and then you must respond to the drag events as they happen by creating an event handler. The following is an example of a Delphi form that accepts dropped files and adds the names of the files to a memo component:

unit Unit ;
interface
uses
  Windows, Messages, SysUtils, Classes, Graphics, Controls, Forms,
  Dialogs, StdCtrls;
type
  TForm  = class(TForm)
    Memo : TMemo;
    procedure FormCreate(Sender: TObject);
  private
    procedure WMDROPFILES(var Message: TWMDROPFILES);
      message WM_DROPFILES;
    { Private declarations }
  public
    { Public declarations }
  end;
var
  Form : TForm ;
implementation
{$R *.DFM}
uses ShellApi;
procedure TForm .FormCreate(Sender: TObject);
begin
 {Let Windows know we accept dropped files}
  DragAcceptFiles(Form .Handle, True);
end;
procedure TForm .WMDROPFILES(var Message: TWMDROPFILES);
var
  NumFiles : longint;
  i : longint;
  buffer : array[0..255] of char;
begin
 {How many files are being dropped}
  NumFiles := DragQueryFile(Message.Drop,
                            - ,
                            nil,
                            0);
 {Accept the dropped files}
  for i := 0 to (NumFiles -  ) do begin
    DragQueryFile(Message.Drop,
                  i,
                  @buffer,
                  sizeof(buffer));
    Form .Memo .Lines.Add(buffer);
  end;
end;
end.