First you need to find out the RSS feed of your website (most websites have that feature).
Then, the best option is the FeedBurner email subscriber widget available at http://feedburner.google.com.
You'll need to create an account or sign in using your Google account and provide Feedburner with the RSS feed address for your blog. You can then click on the Publicize tab at the top of the page and select E-mail subscriptions on the left. Copy the code Feedburner provides and paste it into your website as HTML code.
Taken from https://getsatisfaction.com/weebly/topics/setting_up_a_weebly_blog_for_email_updates