Does my Digitalocean account come with an email server setup?

No. But you can setup your own mail server or subscribe to Google Apps if you have the budget, if you are on the low side, you may try Zoho Apps Email. It’s free for 1 domain per account, and up to 25 users/emails.

Outsourcing your email to Google or Zoho will minimize the load on your server, except that if you are actually planning on setting up another droplet just for the mail server.

I personally use Zoho on my website.

You don’t have to setup an email server in order to have an email @yourdomain.com. You can create a relay via mailgun.com.

Mailgun allows you to create infinite addresses under your own domain, and it provide SMTP server + credential where you can use to establish a secure connection between mailgun and gmail/hotmail inbox. All for free, until you have sent over 10,000 emails a month then you will have to pay $5 for additional 10,000 emails. This does not apply to most of us however.

So once done correctly, you can send and receive under example@mysite.com directly from Gmail inbox as if you run your own email server for example@mysite.com. Here’s full guide.