Signup page in HTML/PHP
I see multiple instances of id="form_input". These attributes need to change to class or each id attribute value should be altered to be unique.
Instead of using &lt;br&gt; tags to style your input fields on separate lines, use css.  By setting all of the elements in the form as display:block;, you can eliminate the excess markup -- a cleaner alternative.  CSS should be the first technique to create spacing and layout stylings; when you find yourself creating spaces with the spacebar or with &lt;br&gt;, try to achieve the same effect with display, margin, padding, etc.
11 reasons why placeholders are problematic -- I can appreciate that you intend to design a clean, minimalist layout, but I don't think I support non-labeled fields and your project should maintain a consistent theme considering that your project may eventually need to employ more forms.
In addition to the previous bullet point about UX, I will urge you to add some client-side validation to your form.  These character limitations will be up to you.  I assume empties.js is doing something that enables the submit button, but I can't see it.  If empties.js is as functionally narrow a file as it's name suggests, you may want to build more robust processes in there.
Rather than sanitizing server-side and unconditionally inserting into the database, you should be catching overtly "bad" submissions and denying access.
If a submitted password contains html markup, that is no concern of yours (no need to sanitize) because you will never, ever be printing that value -- ever.  More clearly: never alter someone's submitted password characters.
You must never, ever present the raw error message to the end user.  See @YCS's hyperlink.  Especially with registration forms, you need to find the correct balance between informing the registrant why the submission was denied, yet not compromising the security of your project.
Finally, login/registration systems are widely available now.  If you are doing this as an educational exercise, then carry on.  If you don't want to reinvent the wheel, just use a pre-existing script from a security-minded author.