This new version of ASP.NET is quite a revolution in the Microsoft ecosystem.
Before ASP.NET Core, ASP.NET was not open source, and ran only on Windows. To develop ASP.NET applications, most developers used Visual Studio as IDE. It was not mandatory, because we could use Notepad with CSharp or VB compilers (which were always free), but so much easier with VS even with an express edition. At that time, the only way to run .NET applications on Linux and macOS was to use Mono, an open source cross-platform version of the .NET Framework.
The ASP.NET Core version 2.0 is far more flexible. You can develop on any OS (at least Windows, Linux, and macOS), use various tools such as Visual Studio, Visual Studio Code, or even Sublime Text. ASP.NET Core is now modular, more maintenable, and has increased performance. By design, it is cloud-ready and middleware-based.
With the new version of ASP.NET, IIS is not the only possible host. You can host your ASP.NET applications on other web servers, such as Kestrel on macOS and Linux.
Among all the changes, some of the most important were the fusion of MVC and Web API, and the deletion of WebForms, Service locator, and System.Web.
All the duplicate libraries existing both in MVC and Web API, such as Controllers, Actions, Filters, Model Binding, and Dependency Resolver, are now the same unified classes.
The strong dependency on the heaviest library used in ASP.NET, System.Web, is now deleted to make ASP.NET Core MVC more modular, middleware-compliant, and platform-independent.