You need to set the AllowsTransparency="True" Popup Property to True
Here is an example:
<Grid>
<StackPanel>
<Button Click="Button_Click" Width="100" Height="20" Content="Click" />
<Popup x:Name="popup" Width="100" Height="100" AllowsTransparency="True">
<Grid Background="Transparent">
<TextBlock Text="Some Text" />
</Grid>
</Popup>
</StackPanel>
</Grid>
and the click handler
private void Button_Click(object sender, RoutedEventArgs e)
{
popup.Visibility = System.Windows.Visibility.Visible;
popup.IsOpen = true;
}