.NET is a general purpose development platform. It has several key features that are attractive to many developers, including automatic memory management and modern programming languages, that make it easier to efficiently build high-quality apps. Multiple implementations of .NET are available, based on open .NET Standards that specify the fundamentals of the platform.
You can read more about the general .NET Framework in the .NET Primer.
.NET Core is a cross-platform implementation of the .NET Framework that is primarily being driven by ASP.NET 5 workloads, but also by the need and desire to have a pay-as-you-go modern runtime that you can customize . You can learn more about .NET Core and how and where you can use it in the CoreCLR is open source blog post.
.NET Core consists of the runtime (CoreCLR) and the libraries (CoreFX). Each of these is available on GitHub, either for raising issues or collaborating on development. The main goal is to create a streamlined, performant and cross-platform execution environment for modern applications.
If you want to understand more details about .NET Core and its ecosystem, you can head read the deeper understanding write up that explains the entire stack, what are the moving pieces and so forth. You can also check out the support page to see a great list of resources where members of our team and of wider community can help you.