Hi there!
Yes, you can definitely do this.
You will have to work with Apache (or any other web server [eg:- Nginx]).
Please look up "Virtual Hosts".
I gathered a small list of existing DigitalOcean tutorials that could be of help to you.
Do let me know if you need anything else.

1) https://www.digitalocean.com/community/tutorials/how-to-set-up-multiple-wordpress-sites-on-a-single-ubuntu-vps
This link talks about wordpress sites on an ubuntu server (you havent mentioned what server youre using but this tutorial involves -- apache, ubuntu, wordpress.

2) https://www.digitalocean.com/community/tutorials/how-to-host-multiple-websites-securely-with-nginx-and-php-fpm-on-ubuntu-14-04
This link is the tutorial answer to the question you asked, except it uses Ubuntu and Nginx
Nginx is considered to be for slightly advanced users, so you would wanna check out apache first.

3) https://www.digitalocean.com/community/tutorials/how-to-set-up-apache-virtual-hosts-on-ubuntu-14-04-lts
This link talks to you about setting up apache virtual hosts - as i mentioned earlier.

To conclude, I think that a thorough reading of all the three links should give you a firm answer to your question.
Let me know if you need anything else.

@andrewcrayden

You can definitely setup a Droplet and configure Apache/NGINX/Caddy/[insert-web-server] as your web server and run PHP, NodeJS, Python, Java or [insert-programming-language] as well. The only thing you'd need to keep in mind is the requirements for each.

If you're going to run multiple environments, make sure you have the resources to handle them. I'd recommend at least a 2GB Droplet if this is the plan, especially if you plan on also using a database server (such as MySQL, MariaDB, Percona, Postgres etc).

The vast majority can be setup from the CLI using repository packages and either yum or apt-get (CentOS will use yum while Debian & Ubuntu use apt-get).

--

For example, to install NGINX + PHP-FPM on Debian/Ubuntu with support for MySQL (or any fork of MySQL, such as MariaDB / Percona), you'd issue the following commands from the CLI:

NOTE: Debian does not pre-install sudo, the package must be installed first. This isn't required for Ubuntu. So if you choose Debian, you'd issue the first command to install the sudo package first and then proceed with the remaining commands. If you choose Ubuntu, simply omit the first command.

Installs sudo:

apt-get install -y sudo

Installs nginx:

sudo apt-get install -y nginx

Installs php-fpm:

sudo apt-get install -y php5-cli php5-fpm php5-dev php5-common php5-mysqlnd

Installs mariadb (a fork of MySQL):

sudo apt-get install mariadb-server

The NGINX configuration file will be located at:

/etc/nginx/nginx.conf

An example server block configuration can be found in:

/etc/nginx/sites-available/

At this point, you would need to configure NGINX to utilize PHP by editing the server block (see the default example in the above directory for a basic guide) and then restarting NGINX:

sudo service nginx restart

And to ensure PHP-FPM is running:

sudo service php5-fpm start

--

Of course, there's more to be done and this isn't an all-inclusive guide. rather, a simple one to show you how to go about getting started without having to manually configure NGINX & PHP (though you can do that as well, it's just a bit more time consuming).

Could it be done with the same domain but having different routes?

<VirtualHost *:80>
    ServerAdmin admin@something.com
    ServerName mydomain.com/project_1
    ServerAlias www.mydomain.com/project_1
    DocumentRoot /var/www/test.com/public_html
    ErrorLog ${APACHE_LOG_DIR}/error.log
    CustomLog ${APACHE_LOG_DIR}/access.log combined
</VirtualHost>

And the other one would be:

<VirtualHost *:80>
    ServerAdmin admin@something.com
    ServerName mydomain.com/project_2
    ServerAlias www.mydomain.com/project_2
    DocumentRoot /var/www/test.com/public_html
    ErrorLog ${APACHE_LOG_DIR}/error.log
    CustomLog ${APACHE_LOG_DIR}/access.log combined
</VirtualHost>

I've tried it and it didn't really work :(