Create a new Ubuntu 16.04 droplet on DigitalOcean. Make sure it has at least 2GB of RAM, and you provide it with your SSH key.
SSH into the server and install Java, which is required for Minecraft to run.
sudo apt-get update && sudo apt-get install default-jdk
java -version
Additionally, install Screen, which will allow us to run the server in the background.
sudo apt-get install screen
Create a directory to install the server files and download the server. (First, get the direct link for the server JAR here: https://minecraft.net/en-us/download/server/ and use it in the command below.
mkdir minecraft
wget -O server.jar https://direct-link-to-server.jar
Edit the server properties file.
vi server.properties
Start a new screen session.
screen -S Server
Run the server.
java -Xmx1024M -Xms1024M -jar minecraft_server.jar nogui
Exit the screen session.
CTRL + A, D