How do I: Pass a DataTable from a Class to the Main program?

Georgio,  one thing at a time,  I have had a better look at your class and have made some mods.

using System;
using System.Collections;
using System.Collections.Generic;
using System.ComponentModel;
using System.Data;
using System.Data.SqlClient;
using System.Text;
using System.Windows.Forms;

//why do you have these here
using System.Reflection;
using System.Drawing;
using System.Drawing.Printing;
using System.Security.Cryptography;
using System.IO;
namespace lizardMarquee
{
    public class ClassDB
    {
        //These are not needed unless there is a specific use for them
        //---------------------------------------------------------------
        public string DBServer;
        public string DBDatabase;
        public string DBLogin_ID;
        public string DBPassword;
        //
        public string DBSelect;
        public string DBInsert;
        public string DBUpdate;
        public string DBDelete;
        public string DBClose;
        //
        public string DBReadRecord;
        public string DBLoadGrid;
        public string DBLoadCombo;
        public string DBLoadList;
        //
        //---------------------------------------------------------------
        //These should be private
        //---------------------------------------------------------------
        private string dBConnect;

        private SqlConnection DBConnection = null;
        private SqlDataReader DBDataReader = null;
        private SqlCommand DBCommand = null;
        private SqlParameter DBParameter = null;
        private SqlDataAdapter DBDataAdapter = null;
        //
        public ClassDB()
        {
            DBCommand = new SqlCommand();
            DBParameter = new SqlParameter();
        }

        public ClassDB(string strCon)
        {
            DBConnect = strCon;
            DBCommand = new SqlCommand();
            DBParameter = new SqlParameter();
            Connection();
        }
        //

        //        public  void Connection(string DBServer, string DBDatabase, string DBLogin_ID, string DBPassword)
        public void Connection()
        {
            //DBConnect = @"Data Source=.\SQLEXPRESS;Initial Catalog=temp;Integrated Security=True";
            /*
                @"Server=" + DBServer + ";"
                + "Database=" + DBDatabase + ";"
                + "User ID=" + DBLogin_ID + ";"
                + "Password=" + DBPassword + ";";
             */
            //
            try
            {
                DBConnection = new SqlConnection(DBConnect);
                DBConnection.Open();
                DBDataAdapter = SetupDataAdapter();
            }
            catch
            {
            }
            finally
            {
            }
        }
        //make this a property of the class and use it to set your connection
        public string DBConnect
        {
            get { return (dBConnect); }
            set { dBConnect = value; }
        }

        public void Connection(string blank)
        {
            try
            {
                DBConnection = new SqlConnection(dBConnect);
                DBConnection.Open();
                SetupDataAdapter();
            }
            catch
            {
            }
            finally
            {
            }
        }
        //
        public void Terminate()
        {
            try
            {
                if (DBCommand != null)
                {
                    DBCommand.Connection.Close();
                    DBCommand.Connection.Dispose();
                }
            }
            catch (Exception Error)
            {
            }
            finally
            {
            }
            //
            try
            {
                if (DBConnection != null)
                {
                    DBConnection.Close();
                    DBConnection.Dispose();
                }
            }
            catch
            {
            }
            finally
            {
            }
        }
        //
        //
        public void LoadGridView(string sDBCommand, DataGridView DBDataGridView)
        {
            Connection();
            DBDataAdapter = SetupDataAdapter();

            DBCommand.CommandText = sDBCommand;                         //these must be in this order
            DBDataAdapter.SelectCommand = DBCommand;                    //
            DBDataAdapter.SelectCommand.Connection = DBConnection;      //

            try
            {
                DataSet DBDataSet = new DataSet();
                DBDataAdapter.Fill(DBDataSet);
                DBDataGridView.DataSource = DBDataSet.Tables[0];
                Terminate();
                //return (DBDataSet);
            }
            catch
            {
                //return null;   this not needed
            }
            finally
            {
            }
        }
        //

        private SqlDataAdapter SetupDataAdapter()
        {
            return new SqlDataAdapter();
        }

        //
   }
}

if you adopt the above all you would need to do in your form is this.

           ClassDB con = new ClassDB(@"Data Source=.\SQLEXPRESS;Initial Catalog=temp;Integrated Security=True");   //instantiate & connect
           con.LoadGridView("SELECT name, Address, FROM Company WHERE Company_id = ' 00000' ORDER BY name", dataGridView);
           con.Terminate();

I have compiled and loaded a grid view with the modified code, it all works so you should be able to do the same by changing just the Initial Catalog = your database name.

ReadRecord is something you would do if you want to manually load a grid view rather than use a DataAdapeter, so these things are for learning when you are more experienced and confident for you wanting to manually load controls.