OLAP Cubes are still around. I work with them every day. For Microsoft, there are two flavors. There's Multi-Dimensional Cubes and Tabular Model cubes.
Tabular is the newer flavor, it has some advantages. Like easier to get
a cube up and running in shorter time, it uses a new language called
DAX, you can import models from Excel Power Pivot and I imagine the
newer Power BI. You can assign permissions to
specific users and groups contained in Active Directory. Basically you
create Roles with specific permissions, then assign users to the Roles.
Create partitions. And schedule refreshes from SQL Server SQL Agent to
process the cube, for example, if you receive
new data throughout the day, the cube will only display new data when
Multi Dimension Cubes have been around before Tabular. They are a bit complex, if you're coming from the traditional T-SQL world. MDX is a beast of a language, I've worked with it for a few years and every time is a challenge. Although some apps like SSRS have Wizards to help out.
Once you have an OLAP Cube, you can build reports off it.
What type of reports can you build based off an OLAP cube? Microsoft
offers Power BI, Excel, SQL Server Reporting Services (SSRS), Performance Point and Power View.
The OLAP Cubes run in a different database than OLTP, called Analysis
Services. It contains the Cube, Dimension tables and Roles. The IDE
used to build cubes is typically the SQL Server Data Tools, also known
as BIDS depending on the version, and you can use Visual Studio 2013 provide
you install the Data Tools add-in.
Many times, you build a Cube based off a Data Warehouse. A Data
Warehouse has Dimension tables and Fact tables. The dimension are
typically things like, Person, Region, Warehouse, Date, (things) to
slice the data with. The Fact tables contain measures consisting
of Sums, Average, Mins, Max, Avergages, they are almost always numbers.
And you can build hierarchies within the Dim tables, for example:
Country ; State ; County ; City
That's how OLAP Cubes work. Abbreviated edition.