I was wondering could a single droplet support multiple sites?

Definitely! You can host as many sites as you want on a single droplet.

If so how can I setup the dns?

Simply point all of your domains/subdomains to your droplet's IP address. Your webserver will take care of the rest.

Should I be looking at Nginx over apache?

It doesn't matter -- you can use whichever you prefer :)

I have no Nginx experience but the idea is to use my droplets as a testing ground to ultimately deploy Laravel sites.

See How To Set Up Apache Virtual Hosts on Ubuntu 14.04 LTS. You'll need to create a VirtualHost for each separate Laravel app you want to deploy.

Then, upload your Laravel app to /var/www/domain.com/public_html, and update DocumentRoot in your VirtualHost config to point to /var/www/domain.com/public_html/public.

Finally, you'll need to allow Laravel to write to /var/www/domain.com/public_html/storage. You can do that by executing the folllowing commands:

sudo chgrp -R www-data /var/www/domain.com/public_html/storage
sudo chmod -R g+w www-data /var/www/domain.com/public_html/storage

Repeat that for every Laravel app you want to host.