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Uses
  ShellAPI;

{...}

ShellExecute(Application.Handle, 'open', 'c:\test.exe', nil, nil, SW_SHOW);                    

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Uses
  Windows;

{...}

function RunProcess(FileName: string): Longword;
var
  StartupInfo: TStartupInfo;
  ProcessInfo: TProcessInformation;
begin
  FillChar(StartupInfo, SizeOf(StartupInfo), #0);
  StartupInfo.cb          := SizeOf(StartupInfo);
  StartupInfo.dwFlags     := STARTF_USESHOWWINDOW or STARTF_FORCEONFEEDBACK;
  StartupInfo.wShowWindow := SW_SHOW;
  if not CreateProcess(nil,
    @Filename[1],
    nil,
    nil,
    False,
    CREATE_NEW_CONSOLE or
    NORMAL_PRIORITY_CLASS,
    nil,
    nil,
    StartupInfo,
    ProcessInfo)
    then
      Result := WAIT_FAILED;
end;