The big question: What exactly is Power BI?
Short answer: A data visualization tool.
Long answer: Though primarily it is a data visualization tool, Power BI is a very powerful self service BI tool. It is made up of multiple components and they all work in tandem to provide you with a wholesome data discovery experience.
The usual flow of data in Power BI is from fetch data from Sources (both online and on cloud) to Power BI Desktop (though the data can be fetched directly into Power BI service too), model the data, transform the data, apply logic, apply calculations and then finally publish the data to Power BI Service for others to see, collaborate or create content on top of it.
To achieve this Power BI makes use of various components such as Power Query, Power BI Desktop, Power BI Service, Gateways, Power Shell to name a few. Please visit the links to get in depth know how on individual components.