static class Program
{
    /// <summary>
    /// The main entry point for the application.
    /// </summary>
    [STAThread]
    static void Main()
    {
        Application.EnableVisualStyles();
        Application.SetCompatibleTextRenderingDefault(false);

        Application.Run(new MyCustomApplicationContext());
    }
}


public class MyCustomApplicationContext : ApplicationContext
{
    private NotifyIcon trayIcon;

    public MyCustomApplicationContext ()
    {
        // Initialize Tray Icon
        trayIcon = new NotifyIcon()
        {
            Icon = Resources.AppIcon,
            ContextMenu = new ContextMenu(new MenuItem[] {
                new MenuItem("Exit", Exit)
            }),
            Visible = true
        };
    }

    void Exit(object sender, EventArgs e)
    {
        // Hide tray icon, otherwise it will remain shown until user mouses over it
        trayIcon.Visible = false;

        Application.Exit();
    }
}

public form1_FormClosing(object sender, EventArgs e)
{
    e.Cancel = true; // Cancelar o fechamento do form
    Hide(); // Ocultar o form
    // use this.WindowState = FormWindowState.Minimized; para minimizar
    notifyIcon.Visible = true; // Mostrar o notify icon
}

1) Acrescentar no form o "notifyIcon";
2) Acrescentar código no evento "notifyIcon1_MouseDoubleClick";
3) Código:
        if (this.WindowState == FormWindowState.Minimized)
        {
            notifyIcon1.Icon = SystemIcons.Application;
            notifyIcon1.BalloonTipText = "Aplicação minimizada";
            notifyIcon1.ShowBalloonTip(1000);
        }

        else if (this.WindowState == FormWindowState.Normal)
        {
            notifyIcon1.BalloonTipText = "Aplicação maximizada";
            notifyIcon1.ShowBalloonTip(1000);
        }

4) Código no botão que minimiza:
    private void button2_Click(object sender, EventArgs e)
    {
        Hide(); // Ocultar o form
        this.WindowState = FormWindowState.Minimized;
    }