Most Content Management Systems (CMSs) have additional bits of programs, plugins, that you can install to add functionality to the site. Such is the case for building a mailing list and sending email. WordPress has their JetPack plugin, Joomla has AcyMailing, and there are many others.

These plugins can send out email when you make a new post to your subscribers. I don't suggest using these. Why? Email management and delivery is getting harder and harder every day, due to the never-ending fight against spammers. Yes, you can run your own mailing list, but there's no guarantee that your email will actually get delivered. If your website is on the same computer with 1,000 other sites (and that's not uncommon), if one of them starts sending out spam, then the various mail providers will mark that machine as sending spam and not accept any more email from it. You and the other 999 innocent websites will be branded as spammers through no fault of your own. It's easy to install email software and run it, it's hard to actually get the email delivered. I strongly suggest using a 3rd party commercial email provider.