How to implement just the wp commenting area on my custom site?

The Twenty Eleven Theme has some decent code comments that explain what’s happening.

Quick and dirty overview:

All comment functionality goes in comments.php. You include it into your template with the aptly named comments_template which does a lot of stuff (sets up the current commentor, ect) other than just including `comments.php

Your comments.php template might look something like this.

<?php
// Is the post password protected?  Maybe you want to deal with that.
if(post_password_required())
{
    // show a message to the users about the password requirement.

    // `return` out of the file, don't show anything else
    return;
}

// display the comments if they exist
if(have_comments())
{
    // maybe a header or something here

    // wp_list_comments (http://codex.wordpress.org/Function_Reference/wp_list_comments) 
    // actually displays the comments -- it will just work if
    // you use it without any arguments
    wp_list_comments();

    // You can also specify a custom callback to display things:
    // wp_list_comments(array('callback' => 'your_comment_callback'));
}

// are comments open? display the comment form!
if(comments_open())
{
    // `comment_form` does all the work. You can customize it
    // with arguments and filters. Like wp_list_comments this
    // will just work.
    // http://codex.wordpress.org/Function_Reference/comment_form
    comment_form();
}
else
{
    // display a message about comments being close for visitors
    // this optional -- personally, I prefer to show no message.
}

Take a look at the Twenty Eleven’s comment callback for some real insight in ways that you can display comments themselves.