In short, a droplet is a virtual server. Digital Ocean doesn't hold back. These are full fledge, production machines that spin up very quickly.

Thing about droplets is to think of them as "disposable". Sure, you could fire up a droplet and keep it around for ever and keep it up like you would any other machine. However, many folks use scripts to automatically create droplets, install software to use for a while, and then tear them down when they're done. Or you could spin up a machine, install a new version of your application, then destroy the old one... there's all sorts of possibilities.

But Digital Ocean has made is super easy to fire up and destroy droplets, so you're only limited by your imagination.

As far as your DNS goes, you should be able to go into GoDaddy's advance DNS settings and set up an entry to point back to Digital Ocean. So let's say you owned a domain "blueberry.com". You could set up a subdomain using the "NS" type:

example     NS    NS1.DIGITALOCEAN.COM

Now example.blueberry.com would point to Digital Ocean's name servers. Now, you'd have to go into Digital Ocean's DNS settings and point example.blueberry.com to one of your droplets.

One thing I have NOT figured out (and would love to know) is if there is a way to dynamically update Digital Ocean's DNS server to point to the right IP address as I bring droplets up and down.

Of course, I may answer myself by saying I could script it myself. :D

A droplet is a Virtual Server. That's just DigitalOcean's cute little nickname for it.

Forwarding the Domain is super easy. Just register the domain using DigitalOcean's super simple Domain Manager tool, add 3 NS records to your domain. The addresses are the following:
ns1.digitalocean.com
ns2.digitalocean.com
ns3.digitalocean.com

Have fun!