Shopify has tight integration with Mail Chimp, which is an excellent service that allows you to create a mailing list and reach out to your customers. MailChimp has multiple pricing tiers, including one that is free and still offers a lot of functionality.

In various locations throughout your theme, you'll notice newsletter sign up forms.  If you wish to use this functionality, you must provide your form action URL in the Global Settings area of your theme settings.  This is a special URL that is custom to your MailChimp account. For instructions on how to find your form action URL, follow this link.