AJAX Register uses language files for its dialogues. To translate it copy the file: language/en-GB/en-GB.com_ajaxregister.ini and rename it to whatever your language code is.
For example, if you are translating it to German you would copy it to: language/de-DE/de-DE.com_ajaxregister.ini. Then translate the strings within the file. Make sure that each string is in one single line and it's best to avoid single quotes ('). To translate the admin part you do the same thing with the file: administrator/language/en-GB/en-GB.com_ajaxregister.ini.
You can also use components like Translations Manager (J1.5) and Localise (J2.5) which can help you translate the files right from your Joomla administration.
To translate the extra fields labels you need to use one word string for the label, like TERMS_OF_SERVICE which you then translate in the language files:
/language/en-GB/en-GB.com_ajaxregister.ini
/language/de-DE/de-DE.com_ajaxregister.ini
etc.
Keep in mind that all files must be saved in UTF-8 encoding.