Note that both pages will have to be .php
The .html page won't know what to do with the PHP code, and it'll just show it as text.
Whenever you have a text box, you can echo out it's name into the value attribute. If it's the first time the form is loaded, the variable will be empty, so the text box will be empty, also. If it's the second time the page is loaded, and the variable is passed to the page with some value, then the box will be filled with that value.
<input type=text name=firstname value="<?php echo $firstname; ?>">
<input type=text name=lastname value="<?php echo $lastname; ?>">
<input type=submit name=submit value=submit>
---John Holmes...