To summarize what happened in the ASP.NET world within the last two years is not an easy task; in short words, we can say that we're undoubtedly facing the most important series of changes in the .NET Framework since the year it came to life. ASP.NET Core 1.0, which came out in Q3 2016, was a complete re-implementation of the ASP.NET we knew; the brand new framework unites all the previous web application technologies, such as MVC, Web API, and web pages, into a single programming module, formerly known as MVC6. The new framework introduces a fully-featured cross-platform component, also known as .NET Core, shipped with a brand new open source .NET Compiler Platform (currently known as Roslyn), a cross-platform runtime (known as CoreCLR), and an improved x64 Just-In-Time compiler (RyuJIT).
The 1.0 final release was shortly followed by ASP.NET Core 1.1 (Q4 2016), which brought some new features and performance enhancements, and also addressed many bugs and compatibility issues affecting the former one.
The third and, at the time of writing, the latest step was taken with ASP.NET Core 2.0, which came out in Q2 2017 as a preview and then in Q3 2017 for the final release; the newer version features among a number of significant improvements with a huge effort to standardize the shared APIs among the .NET Framework, .NET Core, and Xamarin, making it easy for developers to share and reuse code across the whole .NET ecosystem.
ASP.NET 5 was no less than the original name of ASP.NET Core, before the developers chose to rename it to emphasize the fact that it is a complete rewrite. The reasons for that, along with the Microsoft vision about the new product, are further explained in the following Scott Hanselman's blog post that anticipated the changes on Jan 16, 2016:
http://www.hanselman.com/blog/ASPNET5IsDeadIntroducingASPNETCore10AndNETCore10.aspx
For those who don't know, Scott Hanselman is the outreach and community manager for .NET/ASP.NET/IIS/Azure and Visual Studio since 2007.
Additional information regarding the perspective switch is also available in the following article by Jeffrey T. Fritz, Program Manager for Microsoft and NuGet team leader:
https://blogs.msdn.microsoft.com/webdev/2016/02/01/an-update-on-asp-net-core-and-net-core/
As for Web API 2, it was a dedicated framework for building HTTP services returning pure JSON or XML data instead of web pages. Initially born as an alternative to the MVC platform, it has been merged with the latter into the new, general-purpose web application framework known as MVC6, which is now shipped as a separate module of ASP.NET Core.