How can I embed comments plugin to my own website?

Take a look at rendering a comment_form. You can render it anywhere in your page template and there are lots of ways to customize it.

$comments_args = array (
    // change the title of send button
    'label_submit'        => 'Send',
    // change the title of the reply section
    'title_reply'         => 'Leave Your Comment',
    // remove "Text or HTML to be displayed after the set of comment fields"
    'comment_notes_after' => '',
    // redefine your own textarea (the comment body)
    'comment_field'       => '<p class="comment-form-comment"><label for="comment">' . _x( 'Comment', 'noun' ) . '</label><br /><textarea id="comment" name="comment" aria-required="true"></textarea></p>',
);

comment_form( $comments_args );