How to delete seleted item from a listbox using MVVM.

First thing, You need to use a Collection type That will notify the UI of any changes. So this:

public const string MyListPropertyName = "ListBox";
private List<string> _listBox = null;
public List<string> ListBox
{
	get
	{
		return _listBox = _listBox ?? new List<string>();
	}
	set
	{
		if (_listBox == value)
		{
			return;
		}
		RaisePropertyChanged(MyListPropertyName);
		_listBox = value;
		RaisePropertyChanged(MyListPropertyName);
	}
}

can be changed to this:

public ObservableCollection<string> ListBox { get; set; }
    = new ObservableCollection<string>();

Your XAML is missing so I can't see if the SelectedItem property is bound correctly.

I also can not check if your DeleteCommand is enabled correctly. The RelayCommand supports enabling and disabling the bound button. So you could wire it up like:

public RelayCommand DeleteCommand { get; private set; }

private void InitializeRelayCommands()
{
    DeleteCommand  = new RelayCommand
                    (() => DeleteSelectedItemListBox(),
                     () => isSelected);
}

private bool isSelected;

private string selectedItem;
public string SelectedItem
{
    get { return selectedItem; }
    set
    {
        Set(ref selectedItem, value); // ** see below
        isSelected = !string.IsNullOrEmpty(SelectedItem);
    }
}

** The ViewModel needs to inherit the Galasoft ViewModelBase.

Now to delete the item, you only need to add the following where you want to remove the SelectedItem from the collection:

if (!string.IsNullOrEmpty(SelectedItem))
{
    ListBox.Remove(SelectedItem);
    SelectedItem = string.Empty;
}

Lastly, to prevent any confusion, don't call the collection ListBox but instead something meaningful like ContactNames and SelectedItem as SelectedContact. Now your code is self-describing with these two simple changes:

ContactNames.Remove(SelectedContact);

Also, you're using the incorrect namespace for the RelayCommand.Change:

using GalaSoft.MvvmLight.Command;

to:

using GalaSoft.MvvmLight.CommandWpf;

So your MainWindowViewModel now looks like:

using GalaSoft.MvvmLight;
using GalaSoft.MvvmLight.CommandWpf;
using PhoneBook.View;
using System.Collections.Generic;
using System.Data;
using System.IO;
using System.Linq;
 
namespace PhoneBook.ViewModel
{
    class MainWindowViewModel : ViewModelBase
    {
        #region Constructor
        
		public MainWindowViewModel()
        {
			InitializeRelayCommands();
            ReadTextFile();
        }
 
        #endregion
		
        #region Properties
        
        public ObservableCollection<string> ContactNames { get; set; }
            = new ObservableCollection<string>(); 

		private bool isSelected;
		 
		private string selectedContact;
		public string SelectedContact
		{
			get { return selectedContact; }
			set
			{
				Set(ref selectedContact, value);
				isSelected = !string.IsNullOrEmpty(selectedContact);
			}
		}
        #endregion
 
        #region Method
 
        private void PopUpWindSelectedContact()
        {
           
        }
 
        private void DeleteSelectedContactContactNames()
        {
            string FileName = (@"C: \Users\StanleyM\Desktop\PhoneBook\PhoneBook\bin\Debug\Personal.text");
            StreamReader streamReader = new StreamReader(FileName);
            string line = "";
            int Counter = -1;
            
            while ((line = streamReader.ReadLine()) != null)
            {
                foreach (var item in line)
                {
                    if (item == SelectedContact)
                    {
                        Counter--;
                        ContactNames.Remove(line);
                    }
                }
            }
        }
 
        public void ReadTextFile()
        {
            string FileName = (@"C: \Users\StanleyM\Desktop\PhoneBook\PhoneBook\bin\Debug\Personal.text");
            DataTable dt = new DataTable("Client");
            StreamReader streamReader = new StreamReader(FileName);
            string line = "";
            int Counter = 0;
            while ((line = streamReader.ReadLine()) != null)
            {
                Counter++;
                ContactNames.Add(item: line);
            }
        }
 
        private void PopUpWindow()
        {
            AddEditView PopUp = new AddEditView();
            PopUp.ShowDialog();
        }

        #endregion

        #region RelayCommand
 
        public RelayCommand AddCommand { get; private set; }
		public RelayCommand DeleteCommand { get; private set; }
		 
		private void InitializeRelayCommands()
		{
			AddCommand = new RelayCommand
			                (() => PopUpWindow());

			DeleteCommand  = new RelayCommand
							(() => DeleteSelectedContactContactNames(),
							 () => isSelected);
		}

        #endregion
    }
}