I am running Drupal 8 on a DigitalOcean server. I am trying to use the standard Drupal 8 Contact form to get user's email addresses. I currently have the form up and it is visible and holds the information I want but when the user submits their message the form does not send an email to the address that it is set to and two pop ups tell the user that the email is not sent and the message is sent. The message is saved to the contact form storage module that I have downloaded but the email is not sent out. Can anyone help me solve this? I have checked my MX records and the email address that is used in the contact form and is a valid gmail that I have control of and is currently redirected from our server. I am unsure how to solve this issue please help
Most likely you don't have an MTA(Mail Transfer Agent) installed/configured on your droplet. You need to install Postfix so that it can be used to send emails by local applications.
Below is some info from DigitalOcean tutorials library on how to install/setup/test Postfix.
Another option is to use SMTP Authentication Support module
This module allows Drupal to bypass the PHP mail() function and send email directly to an SMTP server. The module supports SMTP authentication and can even connect to servers using SSL if supported by PHP.