The heart of Power BI is a cloud-hosted service for data integration and visualization.
Power BI has a Free Edition with broad functionality, and a Pro Edition, available on a monthly subscription. Power BI is hosted by Microsoft on their "Azure" cloud service.
Power BI is comprised of several tools, web experiences and APIs. Naming of specific components is sometimes confused, so this page attempts to clarify this. These are the main components, with a brief description:
Changelogs can be found here: PowerBI Service, PowerBI Mobile, PowerBI Desktop - Latest Version, Historic Versions
This technique can also be used for other data sources e.g. files, web services.
Note that when you do change these settings, you may have to specify Authentication/Credential details for the new source.
M and DAX are two building blocks of Power BI. They both are functional languages but they differ considerably and are used for very different purposes.
M is used in Power Query (a.k.a. Get & Transform in Excel 2016) and the query tool for Power BI Desktop. Its functions and syntax are very different from Excel worksheet functions. M is a mashup query language used to query a multitude of data sources. It contains commands to transform data and can return the results of the query and transformations to either an Excel table or the Excel or Power BI data model.
More information about M can be found in this MSDN area: Introduction to Power Query (informally known as "M") Formula Language
DAX stands for Data Analysis eXpressions. DAX is the formula language used in Power Pivot and Power BI Desktop. DAX uses functions to work on data that is stored in tables. Some DAX functions are identical to Excel worksheet functions, but DAX has many more functions to summarize, slice and dice complex data scenarios.
There are many tutorials and learning resources for DAX. The Introduction to DAX is a good place to start.
In essence: First you use Power Query (M) to query data sources, clean and load data. Then you use DAX to analyze the data in Power Pivot. Finally, you build pivot tables (Excel) or data visualisations with Power BI.
Power Query and DAX resources can also be found on lots of blogs (e.g. Power BI Team Blog) and in the "Desktop" section of the Power BI Community site.