SMTP is not related to PHP. Typically, your server is going to have a program (usually Sendmail) that handles the delivery of mail in the SMTP protocol. When PHP was installed on your server, they configured PHP to know where that program is so that when you use the mail() function, PHP hands the outgoing message to your SMTP softare (again, usually called Sendmail). So to answer your question, yes, you will need SMTP but it should already be available - I've never seen a copy of PHP that wasn't already connected to the SMTP software so you almost certainly have it available.
I'm not sure exactly what this "class.phpmailer.php PHPmail" library does (maybe some really advanced features) but you won't need it to do a basic confirmation email even if you need HTML formatting or attachments.
To send a regular mail without HTML formatting (which I suggest for compatibility purposes) do this:
mail($customer_email, "This is the subject line",
" ","From:you@yourdomain.com\nReply-To:you@yourdomain.com\n\n
Thank you for registering $customer_name
blah blah blah
");
To send an HTML email (which I hate), do this:
mail($customer_email, "This is the subject line",
" ","From:you@yourdomain.com\nReply-To:you@yourdomain.com\nMIME-Version: 1.0\nContent-Type: text/html\n\n
<html>
Thank you for registering <b>$customer_name</b>
</html>");
Attachments are also possible but they require about 25 lines of code and you probably don't need them at this stage. If you do need them, Google for:
Wanja Hemmerich email attachments php
