For ashopin

button.php

<?php

function YOURMODULE_form($form,&$form_state) {
    // ... the rest of your form ...
    
    $form['email_button'] = array(
        '#type' => 'button',
        '#value' => 'Send Email',
        '#ajax' => array(
            'event' => 'click',
            'callback' => 'YOURMODULE_ajax_send_email',
        ),
    );
    
    // ... goes in here somewhere ...
}

function YOURMODULE_ajax_send_email($form,&$form_state) {
    // do your send-email stuff here
}