Welcome to digitalocean!
If you would like to handle your droplet both domains to the same directory you need to take a few steps:

• First of all you need to get in your DNS manager, here at DO or at your registrar, and redirect an A record to your droplets IP. So for example site.com –> 104.202.30.1
  Do this for both domains. If you would like to create multiple domains/subdomains, just create a CNAME record.

• Once there has been a DNS update, the domains will be redirected to your droplet. After you’ve done the DNS changes, you need to let your server handle the queries from people. It depends what kind of webserver you are using, but with most of the webservers (nginx, apache) you need to change your virtualhost / create a new one.

• In the virtualhost configuration you need to let it handle the domains. In apache thats being done with:

  <VirtualHost *:80>
  ServerName  domain1.com
  ServerAlias www.domain1.com domain2.info www.domain2.info
  </VirtualHost>
  

  In nginx its being handled as following:

  server {
      listen *:80;
      server_name domian1.com www.domian1.com domian2.info www.domian2.info;
      root   /var/www/website;
  }
  

  Hope this is enough information to get you started. If you need any more help in future just ask here or contact me on skype: cryptic-designs