.NET Framework: The Windows-only .NET runtime and libraries that ship as part of Windows and are maintained with the operating system. The latest version is 4.7, included with Windows 10 Creators Update, and can be installed on Windows Server 2008 R2 and upwards.
.NET Core: The cross-platform fork of .NET, which runs on Windows, macOS, and Linux (various distros are supported). Current version is 1.0, with 2.0 in preview and set for release in the third quarter of 2017.
ASP.NET Core: The next generation of the ASP.NET web framework, which (until now) runs either on .NET Framework or .NET Core. The current version is 1.1, with version 2.0 promised at the same time as .NET Core 2.0.
.NET Standard: A specification that defines APIs available across different versions of .NET. Version 1.6 is supported by .NET Core 1.0 and .NET Framework 4.6.1 and higher, and also by Xamarin on iOS and Android. Version 2.0 is supported by .NET Core 2.0 as well as .NET Framework 4.6.1.