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using...

namespace WpfApplication10
{
    /// <summary>
    /// Interaktionslogik für MainWindow.xaml
    /// </summary>
    public partial class MainWindow : Window
    {
        public class Globals
        {
            public static string Connection = "Server=localhost;Database=data;Uid=root;Pwd=;";
        }
        public MainWindow()
        {
            InitializeComponent();
            FillDataGrid();
        }

        private MySqlDataAdapter sqlDataAdapter;
        private DataTable dataTable;
        private void FillDataGrid()
        {
            string ConString = "Server=localhost;Database=data;Uid=root;Pwd=;";
            string CmdString = string.Empty;
            using (MySqlConnection con = new MySqlConnection(ConString))
            {
                CmdString = "SELECT NName,Name,Street,EMail,MobileNo FROM Phonebook";
                MySqlCommand cmd = new MySqlCommand(CmdString, con);
                sqlDataAdapter = new MySqlDataAdapter(cmd); // <--
                dataTable = new DataTable("Phonebook"); // <--
                sqlDataAdapter.Fill(dataTable);
                datagridview1.ItemsSource = dataTable.DefaultView;
            }
        }

        private void UpdateDataGrid()
        {
            sqlDataAdapter.Update(dataTable);
        }

        private void Button_Click(Object sender, EventArgs e)
        {
            DataTable db = new DataTable("Phonebook");
            DataView view = new DataView(db);
            view.RowFilter = "Name LIKE '%" + this.TextBox6.Text + "%'";
            datagridview1.ItemsSource = view;
        }


       
    }
}                    

1:

"Name LIKE '%" + this.TextBox6.Text + "%'" + " OR " + "NName LIKE '%" + this.TextBox6.Text + "%'"; // + " OR " + ...