ASP.NET Core is a free and open-source web framework, and the next generation of ASP.NET, developed by Microsoft and the community. It is a modular framework that runs on both the full .NET Framework, on Windows, and the cross-platform .NET Core.
The framework is a complete rewrite that unites the previously separate ASP.NET MVC and Web API into a single programming model.
Despite being a new framework, built on a new web stack, it does have a high degree of concept compatibility with ASP.NET MVC.
Originally deemed ASP.NET vNext, the framework was going to be called ASP.NET 5 when ready. However, in order to avoid implying it is an update to the existing ASP.NET framework, Microsoft later changed the name to ASP.NET Core at the 1.0 release.