Based on your html source code, it looks like you're using the "Required Fields at Registration" plugin. The file to edit is this one:
oc-content/plugins/requiredreg/form.php
By default, this has the cell phone, user type, and website as fields that are displayed and required. The instructions at the top of the file are "<!-- uncomment those fields you want to use, uncomment next line if you want to use drop-down menus for location -->" but it doesn't remind you that you can also do the reverse, i.e. comment out anything you don't want to use.
To comment the part you don't want to use, you have to add this in front of it:
<!--
and this right after it:
-->
So if you want to comment out the cell phone field, change this:
<label for="phoneMobile"><?php _e('Cell phone', 'modern') ; ?></label>
<?php UserForm::mobile_text() ; ?>
<br />
into this:
<!-- <label for="phoneMobile"><?php _e('Cell phone', 'modern') ; ?></label>
<?php UserForm::mobile_text() ; ?>
<br /> -->
If you want a field to be available at registration but not required, first you uncomment it and then, in the javascript part of the file at the bottom, I think you can just change the "required: true" to "required: false" in the line corresponding to the field you want to use but not require. So for the mobile phone, this line:
$("#s_phone_mobile").rules("add", {required: true, messages: { required: "Mobile phone is required" }});
would get changed to this:
$("#s_phone_mobile").rules("add", {required: false, messages: { required: "Mobile phone is required" }});
Either that or else you have to comment it out like this:
/* $("#s_phone_mobile").rules("add", {required: true, messages: { required: "Mobile phone is required" }}); */
Give it a try and see if it works. If it doesn't, let me know and I'll see what else you could try.
:-)
Marian