Some basic terminology:
Cloud: No such thing. It is just your stuff on someone else’s machines that they maintain for you.
Virtual Machine (VM): A Virtual Server on some physical servers…yours, or someone else’s.
Azure: Fancy name for Microsoft’s cloud. As a noun or an adverb it means “blue”. Or a small butterfly.
Azure SQL database: Just a database in Azure on some storage
Azure Virtual Machine: A VM on Microsoft’s Azure servers, that you do not have to maintain the underlying physical infrastructure.
This is a nice, very high-level introduction to why Azure SQL Database exists.