What is the best way to listen for changes in JTable cell values and update database accordingly?

You can implement the CellEditorListener interface, as shown in this example. Note that JTable itself is a CellEditorListener.

It may also be convenient to terminate the edit when focus is lost, as shown here:

table.putClientProperty("terminateEditOnFocusLost", true);

More Swing client properties may be found here.

I'm agreeing with @mKorbel - unless all your input is checkboxes and dropdowns, you're going to want to wait until the cell editing is stopped (you don't want to commit to the database every time a letter is typed in a textbox).

If the problem is that it's not committing after focus has gone to another component, add a FocusListener that stops editing the table when focus is lost on the table:

Example:

final JTable table = new JTable();
table.addFocusListener(new FocusAdapter() {
    @Override
    public void focusLost(FocusEvent e) {
        TableCellEditor tce = table.getCellEditor();
        if(tce != null)
            tce.stopCellEditing();
    }
});