MySQL Cloud Service is a new(ish) way to run MySQL in the cloud, without worrying about installing the server yourself and configuring security, performance, or monitoring, because that's all taken care of by the service. This makes it great for setting up quick throwaway instances that you can just as easily tear down, and it's ideal for developers who don't want to get in line to wait for their DBAs to set up a server for them.
If you've got an account with Oracle Public Cloud (or even a trial account) that has MySQL Cloud Service enabled, it's only a few clicks to get an instance set up. Here's a short overview.
When you've got your instance, you can connect to it with several different techniques, including command-line access with SSH, MySQL Workbench, or from an application hosted either within or outside Oracle Cloud. For example, here's a tutorial that shows how to create a PHP Rest application that connects to MySQL Cloud Service, without having to install and configure a MySQL server yourself.
You'll find more tutorial videos and walkthroughs for MySQL Cloud Service at the Oracle Help Center.