How to display a message window in the right bottom corner of the active display using Delphi

Try using the TJvDesktopAlert component wich is included in the JVCL, you can find an example in jvcl\examples\JvDesktopAlert\JvDesktopAlertDemo.dpr

_{(source: agnisoft.com)}

unit Unit1;

interface

uses
  Windows, Messages, SysUtils, Variants, Classes, Graphics, Controls, Forms,
  Dialogs, StdCtrls;

type
  TForm1 = class(TForm)
    Button1: TButton;
    procedure FormCreate(Sender: TObject);
    procedure Button1Click(Sender: TObject);
  private
    { Private declarations }
  public
    { Public declarations }
  end;

var
  Form1: TForm1;

implementation

{$R *.dfm}

procedure TForm1.FormCreate(Sender: TObject);
  function TaskBarHeight: integer; // this is just to get the taskbar height to put
  // my form in the correct position
  var
    hTB: HWND;
    TBRect: TRect;
  begin
    hTB := FindWindow('Shell_TrayWnd', '');
    if hTB = 0 then
      Result := 0
    else
    begin
      GetWindowRect(hTB, TBRect);
      Result := TBRect.Bottom - TBRect.Top;
    end;
  end;

begin
  Self.Left := Screen.Width - Self.Width;
  Self.Top := Screen.Height-Self.Height-TaskBarHeight;
end;

procedure TForm1.Button1Click(Sender: TObject);
var
  i: integer;
  TimeSleep: integer;
begin
  TimeSleep := 5; // the speed you want to show/hide.Increase/descrease this to make it faster or slower
  for i := 1 to Self.Height do
  begin
    Self.Top := Self.Top+1;
    Sleep(TimeSleep);
  end;
  // now let's show it again(use this as code as the show code)
  for i := 1 to Self.Height do
  begin
    Self.Top := Self.Top-1;
    Sleep(TimeSleep);
  end;
end;

end.

via http://www.experts-exchange.com/Programming/Languages/Pascal/Delphi/Q_25043483.html

What you are searching for are Balloon Tips in a System Tray. For general WinAPI here's a nice tutorial for it, that you shouldn't have problems translating to Delphi.

You can find some ready to use code for balloon tips in Delphi here.

A nice implementation is available here.