In short: first, there was the Microsoft .NET Framework, which consists of a runtime that executes application and library code, and a nearly fully documented standard class library.
The runtime is the Common Language Runtime, which implements the Common Language Infrastructure, works with The JIT compiler to run the CIL (formerly MSIL) bytecode.
Microsoft’s specification and implementation of .NET were, given its history and purpose, very Windows- and IIS-centered and “fat”. There are variations with fewer libraries, namespaces and types, but few of them were useful for web or desktop development or are troublesome to port from a legal standpoint.
So in order to provide a non-Microsoft version of .NET, which could run on non-Windows machines, an alternative had to be developed. Not only the runtime has to be ported for that, but also the entire Framework Class Library to become well-adopted. On top of that, to be fully independent from Microsoft, a compiler for the most commonly used languages will be required.
Mono is one of few, if not the only alternative implementation of the runtime, which runs on various OSes besides Windows, almost all namespaces from the Framework Class Library as of .NET 4.5 and a VB and C# compiler.
Enter .NET Core: an open-source implementation of the runtime, and a minimal base class library. All additional functionality is delivered through NuGet packages, deploying the specific runtime, framework libraries and third-party packages with the application itself.
ASP.NET Core is a new version of MVC and WebAPI, bundled together with a thin HTTP server abstraction, that runs on the .NET Core runtime – but also on the .NET Framework.