By default, the MongoDB listens on the local interface. In order to allow your Node application to access the MongoDB instance, modify the value of bindIp in /etc/mongod.conf If you do so, you are highly advised to first review the security checklist from the MongoDB documentation.

The relevant section of the config file should look like:

# [ ... ]
net:
  port: 27017
  bindIp: 127.0.0.1,ip_of_mongo_droplet
# [ ... ]

In addition to enabling one of the forms of authentication supported by MongoDB, you should set up a firewall to limit access to the MongoDB instance. You can do this with a DigitalOcean firewall or UFW which comes pre-installed on Ubuntu Droplets.

If you go with UFW, this command will allow access from the Node app's IP address and nowhere else:

sudo ufw allow from node_ip_address to any port 27017

Now that MongoDB is listening for connections from the outside and has been secured with a firewall, you can connect to it from Node using the MongoDB Node.JS driver

const MongoClient = require('mongodb').MongoClient;
const assert = require('assert');

// Connection URL
const url = 'mongodb://mongo_ip_address:27017';

// Database Name
const dbName = 'myproject';

// Use connect method to connect to the server
MongoClient.connect(url, function(err, client) {
  assert.equal(null, err);
  console.log("Connected successfully to server");

  const db = client.db(dbName);

  client.close();
});

In order to make your app more portable, consider making the value of url in the above example configurable using an environment variable.