Many plugins currently available allow you to enable this feature on your site. My favorite is Jetpack Subscriptions for reasons including:
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Very much up-to-date, well-maintained plugin
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Emails sent from WordPress.com servers, so there’s absolutely no load on your servers
(Make sure you disable all the unnecessary modules activated by default when you activate Jetpack by WordPress.com plugin.)
If you want the emails to be sent from your servers, I think you’d be better off using Subscribe to Comments or Subscribe To Comments Reloaded—whichever suits you best.