How do I make a splash screen for my application?

First make a new form, this will be your SpashSreen.

Set Name to "Splash".
Set BorderStyle to "bsNone".
Put an image or whatever on it.
Make sure it is not auto-created. (Shift-Ctrl-F11)

Now edit your main program body:

program MyApp;
{... }
begin
Application.Initialize;
{ ---------- PUT THIS IN: ------------- }
Splash := TSplash.Create(Application);
Splash.Show;
Splash.Refresh;
{ ------------------------------------- }
..
Application.CreateForm(...);
Application.Run;
end;

Now edit the OnShow event of your main form:

procedure TMainForm.FormShow(Sender: TObject);
begin

{...}

{ Free Splash screen }
Splash.Free;
end;

You now have a splash screen!
Tip: If you place the Spash.Free in a OnTimer event, you can control how long the user sees your splash screen.