Note: your system probably requires you to use 'sudo' in front of your privilege access commands, or need to issue 'su root' to gain full root access. Try

sudo cat /etc/resolve.conf 

So, reboot cleared out your DNS cache and now your DNS servers have gone dark. Can you ping IP address 8.8.8.8 or 8.8.4.4 ?

I always log in with root, but tried sudo cat /etc/resolve.conf anyway. The result is the same "No such file or directory".

I can ping both 8.8.8.8 and 8.8.4.4. I can also ping the gateway.

One additional note though, I was playing around with ufw before restarting the droplet. However, after discovering this issue, I have inactivated it, but the problem persists.

@medowlock - Does your system have a /var/run/resolvconf/resolv.conf
If it's there, for diagnostic purpose, create a link to it in etc

ln -s /var/run/resolvconf/resolv.conf /etc/resolv.conf

Then try ping digitalocean.com

This is where it gets really weird, at least for me. When listing the contents of /etc I can see there is a file resolv.conf symlinked to /run/resolvconf/resolv.conf. However

root@ubuntu-512mb-fra1-01:/run# cd /run/resolvconf
-bash: cd: /run/resolvconf: No such file or directory

And running the command you mentioned gives

root@ubuntu-512mb-fra1-01:~# ln -s /var/run/resolvconf/resolv.conf /etc/resolv.conf
ln: failed to create symbolic link ‘/etc/resolv.conf’: File exists
root@ubuntu-512mb-fra1-01:~# cat /etc/resolv.conf
cat: /etc/resolv.conf: No such file or directory

root@ubuntu-512mb-fra1-01:~# ping digitalocean.com
ping: unknown host digitalocean.com

Thanks for all your answers and patience!

@medowlock - Oops, need to remove the existing file before the link command:

mv /etc/resolv.conf /etc/resolv.conf.hold
ln -s /var/run/resolvconf/resolv.conf /etc/resolv.conf