Cloud or Cloud Computing is a general term that means delivering storage, compute, network, application, software as a complete stack or one or more of them over the internet (which is the back bone of cloud).
Companies offering these computing services are called Cloud Providers and typically charge for cloud computing services based on usage, just like how you are billed for utilities like water or electricity at home.
Preparing a spreadsheet over the Internet using Google Sheets is a simple example of cloud computing. Instead of using a program like Microsoft Excel which is running on your computer, you use a software running on a PC at one of Google’s world-wide data centers.
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A database is a set of data, collection of information that has a regular structure and is organized so that it can easily be accessed, managed on a computer and thereby making it easy for you to find the desired information.
Now you can have a Cloud Database which is a database that typically runs on a Cloud Computing platform.