You will need to use Delphi's FindFirst() and FindNext() procedures to create a list of all the sub directories in a directory. For each sub directory, you will need to use Delphi's FindFirst() and FindNext() procedures, again to add any sub directories of the found directories to the list. Continue working down the list until all the directories have been processed. Example:

procedure GetDirectories(const DirStr : string;
                         ListBox : TListBox);
var
  DirInfo: TSearchRec;
  r : Integer;
begin
  r := FindFirst(DirStr + '\*.*', FaDirectory, DirInfo);
  while r = 0 do  begin
    Application.ProcessMessages;
    if ((DirInfo.Attr and FaDirectory = FaDirectory) and
         (DirInfo.Name <> '.') and
         (DirInfo.Name <> '..'))  then
      ListBox.Items.Add(DirStr + '\' + DirInfo.Name);
    r := FindNext(DirInfo);
  end;
  SysUtils.FindClose(DirInfo);
end;
procedure GetFiles(const DirStr : string;
                   ListBox : TListBox);
var
  DirInfo: TSearchRec;
  r : Integer;
begin
  r := FindFirst(DirStr + '\*.*', FaAnyfile, DirInfo);
  while r = 0 do  begin
    Application.ProcessMessages;
    if ((DirInfo.Attr and FaDirectory <> FaDirectory) and
        (DirInfo.Attr and FaVolumeId <> FaVolumeID)) then
      ListBox.Items.Add(DirStr + '\' + DirInfo.Name);
    r := FindNext(DirInfo);
  end;
  SysUtils.FindClose(DirInfo);
end;
procedure TForm .FormCreate(Sender: TObject);
var
  i : integer;
begin
  ListBox .Items.Clear;
  ListBox2.Items.Clear;
  ListBox .Items.Add('C:\Delphi');
  GetDirectories('C:\Delphi', ListBox );
  i :=  ;
  while i < ListBox .Items.Count do begin
    GetDirectories(ListBox .Items[i], ListBox );
    Inc(i);
  end;
end;
procedure TForm .ListBox Click(Sender: TObject);
begin
  ListBox2.Clear;
  GetFiles(ListBox .Items[ListBox .ItemIndex],
           ListBox2);
end;
Note: Recursing directories can take up a lot of memory. It is suggested that you consider creating a linked list or a temporary file to store your list entries instead of using a component (such as a Memo or StringList), as there are limits to the number of entries these components can have.