<DataTemplate x:Key="postBody" TargetType="{x:Type Post}">
  <Grid>
    <TextBlock Text="{Binding Path=author}"/>
    <Button Click="btnDelete_Click" Content="Delete"/>
  </Grid>
</DataTemplate>
<ListBox ItemTemplate="{StaticResource postBody}" ItemSource="{Binding Posts}"/>

Code behind to get a reference to object owning btnDelete:

private void btnDelete_Click(object sender, RoutedEventArgs e)
{
  var post = ((Button)sender).DataContext as Post;
  if (post == null)
  {
    throw new InvalidOperationException("Invalid DataContext");
  }
 
  Console.WriteLine(post.author);
}

To find elements generated by a DataTemplate:

<DataTemplate x:Key="myDataTemplate">
  <TextBlock Name="textBlock" FontSize="14" Foreground="Blue">
    <TextBlock.Text>
      <Binding XPath="Title"/>
    </TextBlock.Text>
  </TextBlock>
</DataTemplate>
...
<ListBox Name="myListBox" ItemTemplate="{StaticResource myDataTemplate}"
         IsSynchronizedWithCurrentItem="True">
  <ListBox.ItemsSource>
    <Binding Source="{StaticResource InventoryData}" XPath="Books/Book"/>
  </ListBox.ItemsSource>
</ListBox>

We have to find the ContentPresenter first:

// Getting the currently selected ListBoxItem 
// Note that the ListBox must have 
// IsSynchronizedWithCurrentItem set to True for this to work
ListBoxItem myListBoxItem =
    (ListBoxItem)(myListBox.ItemContainerGenerator.ContainerFromItem(myListBox.Items.CurrentItem));
 
// Getting the ContentPresenter of myListBoxItem
ContentPresenter myContentPresenter = FindVisualChild<ContentPresenter>(myListBoxItem);
 
// Finding textBlock from the DataTemplate that is set on that ContentPresenter
DataTemplate myDataTemplate = myContentPresenter.ContentTemplate;
TextBlock myTextBlock = (TextBlock)myDataTemplate.FindName("textBlock", myContentPresenter);
 
// Do something to the DataTemplate-generated TextBlock
MessageBox.Show("The text of the TextBlock of the selected list item: "
    + myTextBlock.Text);
. . .
private childItem FindVisualChild<childItem>(DependencyObject obj)
    where childItem : DependencyObject
{
    for (int i = 0; i < VisualTreeHelper.GetChildrenCount(obj); i++)
    {
        DependencyObject child = VisualTreeHelper.GetChild(obj, i);
        if (child != null && child is childItem)
            return (childItem)child;
        else
        {
            childItem childOfChild = FindVisualChild<childItem>(child);
            if (childOfChild != null)
                return childOfChild;
        }
    }
    return null;
}

Source: MSDN: FrameworkTemplate.FindName Method

For a ListBoxItem:

<ListBox Name="listBox">
    <ListBox.ItemTemplate>
        <DataTemplate DataType="x:Type local:NumericIconDefinition">
           <Grid>
                <ComboBox Name="IconComboBox"/>
           </Grid>
        </DataTemplate>
    </ListBox.ItemTemplate>
</ListBox>

ListBoxItem lbi = (ListBoxItem)listBox.ItemContainerGenerator.ContainerFromIndex(IndexInListBox);
ComboBox cb = (ComboBox)listBox.ItemTemplate.FindName("IconComboBox", lbi);

For a ListBoxItem:

<Window.Resources>
    <local:GreekGods x:Key="greekGods"/>
    <DataTemplate x:Key="itemTemplate">
        <TextBlock Text="{Binding Path=Name}" />
    </DataTemplate>
</Window.Resources>
. . .    
<ListBox ItemsSource="{StaticResource greekGods}" ItemTemplate="{StaticResource itemTemplate}" Name="listBox"/>

// To get GreekGod object
GreekGod greekGod = (GreekGod)(listBox.Items[0]);
 
// To get ListBoxItem container holding object GreekGod
ListBoxItem lbi1 = (ListBoxItem)(listBox.ItemContainerGenerator.ContainerFromIndex(0));
ListBoxItem lbi2 = (ListBoxItem)(listBox.ItemContainerGenerator.ContainerFromItem(listBox.Items.CurrentItem));

Source: ZAG Log: How to get a ListBoxItem from a data bound ListBox