New Sites does not have a built-in commenting system at this time. Use of a Google Form and Spreadsheet is demonstrated here as a workaround.
See other methods in the subpages such as GraphComment, Disqus and Email Me
Feel free to use this form to send comment, but if you need help please use the help forum: https://support.google.com/sites/community/
You may want to call it a "Contact me" form. Whatever you use it for, when someone submits the form you get notified, then you access the associated spreadsheet to read the submission content or if you want to you can embed the spreadsheet in your page as I have done here.
See Steegle's instructions which includes a video tutorial and a method to display the most recent comment first.
Note that if you hover over the form or the spreadsheet a pop-out icon appears. It is not necessary to click this as you can fill out the form directly on this site. If you do pop out the form or spreadsheet you will see a link to request to edit it. Please do not do that.
Steps I used to make this Contact Form
Under Google Docs, Click Create New.
Create the form questions that you want.
Click RESPONSES and click the spreadsheet icon.
A spreadsheet will be created that will store the form responses. Open that spreadsheet, Click the Tools menu and select "Notification rules...".
Create a rule. Under "Notify me at [your address] when..." put a check in "A user submits a form" and either "Email - daily digest" or "Email - right away". See 'Notification to Multiple Addresses' below.
Save the rule.
Go to your site and on the page where you want the form. In the Insert menu click Forms in the Google Docs section.
Select the form you just created.
Save the page.
Notice that under step 6 you will be notified that someone has submitted your form, and then you have to open up the spreadsheet to see what they entered.
If you want to use this as a comment form where the comments are posted on your site click Share while editing the spreadsheet and select to share it to the public and on the File menu select Publish to the Web and to automatically re-publish when changed** and click Publish as a web page. Insert the spreadsheet itself as shown below. In the INSERT menu select Sheets in the Google Docs section and select the spreadsheet associated with your form.
Note that the spreadsheet will not automatically update. A viewer will have to refresh the page in order to see new comments and there may be a delay of up to 5 minutes before the new comment appears.
*If you want to moderate the comments do not select automatically re-publish, When you get notified someone has posted you can the publish it. File, Publish to the web, Republish now.
Note that when you create a Form and view it in PREVIEW mode, you will see your email showing at the top of the page. When your respondents view your Form, they will see this same display, but it will show their account address, showing them which account they are currently using as the fill out your Form. They will NOT see your account email.
Notification to Multiple Addresses
Notifications normally go the owner's email. Here are suggestions for sending to multiple people.
In Gmail, set up a filter to forward to the other account(s) or add this add-on.
Add-ons in Forms have features to send notifications to multiple emails. From the form edit view, click the form's 3-dots menu, Add-ons, search for notifications.
This add-on allows users to automate email notifications for:
1. Form respondents - Configure messages that are delivered as individuals submit the Forms submission
2. Form owners and editors - Configure messages that are delivered to you (as the creator of the Form), or other collaborators, on predetermined response thresholds.