Adding column with header and data to the MyTableModel

Question 1

You don't provide any code so I can only guess that your FIRST_COL_VALUES etc are not initialized (perhaps an invalid length?) or perhaps you are replacing the model somewhere else in your code. Displaying 3 columns with data is as simple as can be seen below.

Note that this is a runnable example. If you provide a similar one for your code I would be able to explain in more detail what your error actually is.

package gui.table;

import java.awt.Dimension;
import javax.swing.JFrame;
import javax.swing.JScrollPane;
import javax.swing.JTable;
import javax.swing.SwingUtilities;
import javax.swing.table.DefaultTableModel;

public class ThreeColumnDemo {

    public static class ThreeColumnModel extends DefaultTableModel {

        public ThreeColumnModel() {
            // Initialize with 3 rows 0 columns
            super(3, 0);

            // add columns with data for each row
            this.addColumn("First Column", new String[] {"Value1_1", "Value2_1", "Value3_1" });
            this.addColumn("Second Column", new String[] {"Value1_2", "Value2_2", "Value3_2" });
            this.addColumn("Third Column", new String[] {"Value1_3", "Value2_3", "Value3_3" });
        }
    }

    public static void main(String[] args) {
        SwingUtilities.invokeLater( new Runnable() {

            @Override
            public void run() {
                JFrame window = new JFrame("3-column demo");
                window.setPreferredSize(new Dimension(600, 400));
                window.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);

                JTable table = new JTable( new ThreeColumnModel() );

                // put the table in a scroll pane so column headers are visible
                window.add(new JScrollPane(table));
                window.pack();
                window.setLocationRelativeTo(null);
                window.setVisible(true);
            }
        });
    }
}

Update 1: You are not correctly extending the DefaultTableModel. When you call addColumn(...) during constructor, the parent inserts a new column name and while it tries to make room for the rows/columns it calls your methods like getColumnCount(), getColumnName() etc which you override. However your model does not use the underlying data of the parent but checks a new structure (an ArrayList) which is empty at the moment. So the parent object encounters incosistent behaviour (i.e. it just added a new column and getColumnCount() returns 0). Just comment out the following methods and your code will work: getColumnCount(), getColumnName(), getRowCount(), getValueAt().

Then you should decide what you want to do. DefaultTableModel has a data structure to store data. If you extend it use those structures and do not overide methods that perform the same behaviour with the parent just for the shake of it. You should extend it if you want to add some additional behaviour.

If on the other hand you want to use your own data structure and your own way to add/remove columns and rows then just extend AbstractTableModel.

Update 2: Regarding the AbstractTableModel and your questions: a model is what its name says. A way to model your data. You do the same thing when you use an Object[] or an array of arrays, or an ArrayList<Object> as in your case. That is a model but rather poor. The table model encapsulates an abstract data structure that can be anything that matches your needs no matter how complex (i.e. the JTree uses it to encapsulate a hierarchical structure of tree nodes).
All Swing components behave the same way: you update your model and use events (usually calling a fireXXX()) to notify the corresponding component to redraw itself.
So in your case if you need to add/remove columns then you should have a method addColumn/removeColumn that would be called externally i.e. after you create your table. The model will update its data internally and call the appropriate fireXXX methods. Regarding what data type you should use, use whatever is more convenient for you. In your example you are using Integer for data type so an Object[] or an Integer[] or the corresponding ArrayList make sense. If you use a more complicated data type you can even introduce a new Column class that would encapsulate the data. It can be as complicated as you need it to be.

Question 2

I created a class MyTableModel which extends DefaultTableModel. And what I want is, to have already initialized three columns with data. Inside a constructor of MyTableModel I set header/data values by calling this.addColumn("FIRST COL", FIRST_COL_VALUES); but I get this exception "Exception in thread "main" java.lang.ArrayIndexOutOfBoundsException: 0 >= 0", how do I fix this?

this is possible with override TableColumn

there are (with quite usefull methods, remove if isn't used)

TableColumn column = new TableColumn();
column.setCellRenderer(....);
column.setModelIndex(int);
column.setHeaderValue("String value");
column.setPreferredWidth(int);
columnModel.addColumn(column);

column = new TableColumn();
column.setCellRenderer(....);
column.setModelIndex(int);
column.setHeaderValue("String value");
column.setPreferredWidth(int);
columnModel.addColumn(column);

column = new TableColumn();
column.setCellRenderer(....);
column.setModelIndex(int);
column.setHeaderValue("String value");
column.setPreferredWidth(int);
columnModel.addColumn(column);

table.setColumnModel(columnModel);

EDIT, You could start by providing an SCCCE by @MadProgrammer, then for future readers only

import java.awt.*;
import javax.swing.*;
import javax.swing.table.DefaultTableModel;

public class CheckBoxTable extends JPanel {

    public static final String[] COLUMNS = {"Purchased", "Item"};
    public static final String[] INITIAL_ITEMS = {"Milk", "Flour", "Rice", "Cooking Oil", "Vinegar"};
    private static final long serialVersionUID = 1L;
    private CheckBoxDefaultTableModel model = new CheckBoxDefaultTableModel(COLUMNS, 0, 0);
    private JTable table = new JTable(model);

    public CheckBoxTable() {
        setLayout(new BorderLayout(5, 5));
        add(new JScrollPane(table), BorderLayout.CENTER);
        for (int i = 0; i < INITIAL_ITEMS.length; i++) {
            Object[] row = {Boolean.FALSE, INITIAL_ITEMS[i]};
            model.addRow(row);
        }
    }

    private static void createAndShowUI() {
        JFrame frame = new JFrame("CheckBoxTable");
        frame.getContentPane().add(new CheckBoxTable());
        frame.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
        frame.pack();
        frame.setLocationRelativeTo(null);
        frame.setVisible(true);
    }

    public static void main(String[] args) {
        java.awt.EventQueue.invokeLater(new Runnable() {

            @Override
            public void run() {
                createAndShowUI();
            }
        });
    }
}

class CheckBoxDefaultTableModel extends DefaultTableModel {

    private static final long serialVersionUID = 1L;
    private int checkColumn;

    CheckBoxDefaultTableModel(Object[] columnNames, int rowCount, int checkColumn) {
        super(columnNames, rowCount);
        this.checkColumn = checkColumn;
    }

    @Override
    public Class<?> getColumnClass(int columnNumber) {
        if (columnNumber == checkColumn) {
            return Boolean.class;
        }
        return super.getColumnClass(columnNumber);
    }
}

Question 3

Inside the MyTableModel constructor, try calling this line before anything else:

super(ROW_COUNT, COLUMN_COUNT);

where ROW_COUNT and COLUMN_COUNT are the number of rows and columns, respectively, in your table.