WordPress and Email
For basic functionality (new user, lost password, comment moderation) your WordPress instance needs to be able to send email.
Google Apps
You can setup WordPress to handle your email through Google Apps (GSuite) with the gmail-smtp plugin. This works whether you have a single GMail address, or the full GSuite for your domain. One nice advantage of using this method is that DKIM is handled by Google automatically.
MailGun
MailGun is a better delivery service provider in our experience. If you want a fuller feature set for tracking your mail campaigns and so-forth, then MailGun could be the right sender for your needs.
CiviCRM
If you integrate CiviCRM into your WordPress installation, then you might also have configured CiviCRM independently to hand off email to MailGun or another provider through the SMTP configuration. Using MailGun is preferred (over plain 'mail()' or even GSuite) when your WordPress website is being used to send monthly messages (even just a few thousand) to your newsletter and other list subscribers.