ASP.NET Core is a new open-source and cross-platform framework for building modern, cutting edge cloud based internet and intranet applications such as web apps, IoT apps, mobile apps, desktop apps etc.
ASP.NET Core is the next version of ASP.NET 4.5 (earlier known as ASP.NET vNext or ASP.NET 5).
ASP.NET Core runs on .NET Core framework or on the full .NET Framework.
The prime objective of creating ASP.NET Core is to make it light weight (modular with minimum overhead), open source, cross-platform (runs on Windows, Mac and Linux) and fast (much faster than earlier version of ASP.NET).
Being an open source, ASP.NET Core is availabe at GitHub.