how to save checkbox values in database in magento

To make it working well you need to:

1) In your Form class, before $form->setValues($data); add the following:

if (!is_array($data['expertise'])) {
    $data['expertise'] = explode(',', $data['expertise']);
}

2) In your Controller class, before $model->setData($data); add the following:

if (is_array($data['expertise'])) {
    $data['expertise'] = implode(',', $data['expertise']);
}

Moreover, input component of "checkbox" type is not standard in Magento and it works incorrectly in some cases. The best solution here is to use "multiselect" component:

$fieldset->addField(
    'expertise', 'multiselect', array(
    'name'     => 'expertise',
    'label'    => Mage::helper('checkout')->__('Expertise Area'),
    'required' => true,
    'values'   => array(
        array(
            'label' => 'Php',
            'value' => 'php'
        ),
        array(
            'label' => '.Net',
            'value' => 'dotnet'
        ),
    )
));

Please consider the following things.

For example:

if your attribute is expertise Then in your controller saveAction() when saving the checkbox data do

$expertise = isset($your_form_Data['expertise']) ? 1 : 0;

For Grid and Form Page

In your controller you should have Mage::register(...)->getData() or Mage::register(...)

public function editAction()
     ....
     Mage::register('example_data', $model);

On your form _prepareForm()

$model = Mage::registry('example_data'); // NOTE registry('example_data'); NOT registry('example_data')->getData();

$fieldset->addField('entire_range', 'checkbox', array(
      ....
      'checked'    => $model->getExpertise()==1 ? 'true' : 'false',
       ......
))

On your grid _prepareColumns()

$this->addColumn('expertise', array(
    ....
    'type'     => 'checkbox',
    'index'    => 'expertise',
    'values'   => array(1,2),
    'field_name' => 'checkbox_name',
    ....
));