How to close form

You can also close the application:

Application.Exit();

It will end the processes.

You need the actual instance of the WindowSettings that's open, not a new one.

Currently, you are creating a new instance of WindowSettings and calling Close on that. That doesn't do anything because that new instance never has been shown.

Instead, when showing DialogSettingsCancel set the current instance of WindowSettings as the parent.

Something like this:

In WindowSettings:

private void showDialogSettings_Click(object sender, EventArgs e)
{
    var dialogSettingsCancel = new DialogSettingsCancel();
    dialogSettingsCancel.OwningWindowSettings = this;
    dialogSettingsCancel.Show();
}

In DialogSettingsCancel:

public WindowSettings OwningWindowSettings { get; set; }

private void button1_Click(object sender, EventArgs e)
{
    this.Close();
    if(OwningWindowSettings != null)
        OwningWindowSettings.Close();
}

This approach takes into account, that a DialogSettingsCancel could potentially be opened without a WindowsSettings as parent.

If the two are always connected, you should instead use a constructor parameter:

In WindowSettings:

private void showDialogSettings_Click(object sender, EventArgs e)
{
    var dialogSettingsCancel = new DialogSettingsCancel(this);
    dialogSettingsCancel.Show();
}

In DialogSettingsCancel:

WindowSettings _owningWindowSettings;

public DialogSettingsCancel(WindowSettings owningWindowSettings)
{
    if(owningWindowSettings == null)
        throw new ArgumentNullException("owningWindowSettings");

    _owningWindowSettings = owningWindowSettings;
}

private void button1_Click(object sender, EventArgs e)
{
    this.Close();
    _owningWindowSettings.Close();
}

new WindowSettings();

You just closed a brand new instance of the form that wasn't visible in the first place.

You need to close the original instance of the form by accepting it as a constructor parameter and storing it in a field.