A Droplet is a VPS (Virtual Private Server). The Cloud is essentially a more simple term for a network of servers, each of which may do something different than the next.
When it comes to DigitalOcean and the Cloud, Droplets (i.e. VPS's) are hosted on their Cloud, or their network of servers, which allows you to quickly and easily deploy a barebones OS or one of the one-click images to a location of your choosing (i.e. one of their data centers).
VPS's function much like that of a Dedicated Server in that you're provided with a set amount of CPU, RAM, and SSD (Solid State Disk) Storage. The main difference between the two, other than that fact that VPS's are virtualized using KVM (a type of virtualization technology), are that unlike a Dedicated Server, there are others on the same server you are on.
These clients can't access your data or server, of course, but you're not the sole person as would be the case if you were on a Dedicated Server (which is often more expensive due to that fact).
When you deploy a Droplet (VPS), you get to choose an OS -- Ubuntu is one of them -- though there's also Debian, CentOS, CoreOS and FreeBSD. There's no need to be familiar with all of them, just the one you plan on deploying. In most cases, this will be Ubuntu (with 16.04 and 16.10 being the latest).
Some developers do provide integration with DigitalOcean, thus they offer a one-click button that will allow you to provide authentication details on-site and in turn, it'll deploy a Droplet with said software pre-installed. You also have the option of using the one-click images provided by DigitalOcean to simplify setup and configuration.