MetroTabItem is a TabItem with a close button and the four properties around it. It is what MetroTabControl creates for you when you bind ItemsSource, and its style is based on MahApps.Styles.TabItem, so everything on the TabControl styles page applies here too.

<mah:MetroTabControl>
    <mah:MetroTabItem Header="General" CloseButtonEnabled="True">
        <TextBlock Margin="10" Text="First page" />
    </mah:MetroTabItem>
</mah:MetroTabControl>

The button appears on the selected tab

Look at the figure: only Display has a cross, and all three tabs have CloseButtonEnabled="True".

CloseButtonEnabled alone puts the button in the layout but leaves it Hidden — it takes up its space so the header does not jump. Two triggers make it Visible: the tab being selected, and the pointer being over it. That is the usual pattern for closable tabs, and it is why the figure shows one cross rather than three.

The button's size follows HeaderedControlHelper.HeaderFontSize through a converter, so it stays in proportion when you shrink the tab headers.

<mah:MetroTabControl.ItemContainerStyle>
    <Style BasedOn="{StaticResource {x:Type mah:MetroTabItem}}" TargetType="{x:Type mah:MetroTabItem}">
        <Setter Property="CloseButtonEnabled" Value="True" />
    </Style>
</mah:MetroTabControl.ItemContainerStyle>

What the close button does

The button raises no Click you can handle — a CloseTabItemAction behaviour is attached to it, and that runs a fixed sequence:

1. The item's CloseTabCommand is asked whether it can run. CanExecute returning false keeps the tab and ends it there; otherwise the command runs.
2. Then the control's CloseThisTabItem runs, which raises the closing event and, unless a handler says no, either executes the control's CloseTabCommand or removes the tab.

So there are three ways to keep a tab: a CanExecute of false on the item's command, a cancelled closing event on the MetroTabControl, or a CloseTabCommand on the control, which takes the removal over entirely.

<mah:MetroTabItem Header="Report"
                  CloseButtonEnabled="True"
                  CloseTabCommand="{Binding SaveBeforeCloseCommand}"
                  CloseTabCommandParameter="{Binding RelativeSource={RelativeSource Self}, Path=Header}" />

CloseTabCommandParameter does double duty: it is the parameter for the item's command, and the control's CloseTabCommand is handed it as well, falling back to the MetroTabItem itself where the item has none.

The plain TabItem alternative

TabControlHelper has CloseButtonEnabled, CloseTabCommand and CloseTabCommandParameter as attached properties that work on an ordinary TabItem.

Related

MetroTabControl for the closing pipeline seen from the other end, TabControl for the look, and TabControlHelper for the underline and the transition.