It's taken a bit of effort to cut through the hype and get to the meat of what it is, and what it is expected to be. Here's the no-nonsense version, based on a few hours of experimentation:
What .NET Core is today:* A modularized .NET framework whose libraries are acquired through NuGet
* The newest way to do your ASP.NET apps
* A gateway to using .NET Native, for machine-compiled code instead of JIT
* An adoption by MS of many Mono sources and open source in general for .NET
What .NET Core is also today,
but might/should be categorized under a different name depending where you read:* A new project file structure focusing on "kproj" and Project.json
* Aforementioned new structure facilitates targeting any combination of multiple .NET frameworks within one project file (yes!)
* A way to do very nifty console apps which demonstrate the power of above multitargeting, with zero relation to ASP.NET or .NET Native
What .NET Core isn't, today:* A fully functional replacement for current mature .NET Frameworks. .NET Core is missing many support libraries, such as System.Transaction
* A Client-friendly entity. No WPF for .NET Core yet, based on what I read.
What .NET Core promises to be in the future:* The foundation for all upcoming .NET Frameworks
* A unifying force bringing together things like .NET CF, Silverlight, Windows 8 API, etc.
* Cross platform to Linux, Mac OS as well as Windows
What .NET Core promises to be today, but I can't figure the @#$! out:* A way to build "regular" .NET projects as well, but referencing "regular" .NET Framework libs (like System.Transaction... or any of the others) is darned mysterious
* Can utilize PCL and SHPROJ projects. Can't find a way to make SHPROJ works