You configure email notifications and confirmations in Cognito Forms via Submission Settings, which is accessed on the green footer on the Build page or by clicking on the submit region of the form. Here is the help documentation for this feature:
Notifications can be sent to anyone, registered users or not, and you can include multiple email addresses. Confirmations are sent to the individual filling out the form, so are limited to a single email address. You can optionally include all form entry data, include a detailed receipt, and even include form fields in a custom subject and message for your notifications--lots of options.
The WordPress plugin just provides an integrated experience, but is definitely not required. Cognito Forms is a fully hosted service, so nothing actually runs within WordPress, and all emails are processed using Mandrill, an enterprise transactional email service by MailChimp, so email are not processed by your WordPress site.