function PrintWindow(SourceWindow: hwnd; Destination: hdc; nFlags: cardinal): bool; stdcall; external 'user32.dll' name 'PrintWindow';

procedure TForm1.btn1Click(Sender: TObject);
var
  bmp : TBitmap;
  wnd : cardinal;
  rec : TRect;
begin
   wnd := FindWindow(nil, '计算器'); // 查找窗口句柄，这里用计算器演示
   GetWindowRect(wnd, rec); //获取到计算器窗口的矩形坐标
   bmp := TBitmap.Create;
   try
     bmp.Width := rec.Right - rec.Left;
     bmp.Height := rec.Bottom - rec.Top;
     bmp.PixelFormat := pf24bit;
     PrintWindow(wnd, bmp.Canvas.Handle, 0);
     bmp.SaveToFile('test.bmp');
   finally
     bmp.Free;
  end;
end;