Make a form with a input button that calls an AJAX function using onclick function. Then in the AJAX function, use the .open("GET", "php_file_name" + input_fields, true); and .send(null); to call a php file which would do that. You have to obviously pass the info from the forms into a PHP file's address so it can get the data. Then use php to do the rest.
I don't see why you didn't just go with straight forward PHP. ???
After PHP gets the data, how could I put the data in such a way in the responsetext so that I can use javascript to insert a new row into my table? I am inserting data into a form, that form uses ajax to insert data into a database and then I need it to return the new data inserted in a formatted form and then insert it as a new row in the table that's already on the page without refreshing the page.