It's called "core", because it doesn't contain the Windows desktop UI toolkits, like Windows Forms and Windows Presentation Foundation (WPF), and excludes some other Windows-specific functionality.
Additionally, this is the first version of .NET in which apps can be totally self-contained. Previously, you'd need to install the .NET Framework on your machine to run .NET apps. That is no longer a requirement in .NET Core 1.0.
So, it's a pretty huge release. And really the first time in decades that MS has built it's premier dev stack with multiple OS support.
[0]: https://github.com/dotnet/core/blob/master/release-notes/1.0...
Though I'm not sure it's of any value outside of Windows.
I tried it - it puts 50MB of DLLs beside the "HelloWorld" executable in a subdirectory (on Ubuntu 16.04).
It's like shipping the entire Java runtime with every app.
That's a bad test for a framework. It's like saying, "I bought an $80,000 car to drive to my neighbor's house across the street - I could have just walked!"
A good test is, "What is the full install size of a real app built on .NET Core?"
And the answer is, not much larger than any other app. All that code has to live somewhere, either in your app or in your libraries.
This is a different (but related) product to the .NET Framework (current version, 4.6) -- future versions of the .NET Framework will be built on top of .NET Core, which will be updated more frequently than the larger Framework. So, maybe .NET Framework 5.0 will be built on something like .NET Core 1.3 -- or 2.0, whatever.
So Microsoft would have: .NET Core - base for everything. .NET - Windows specific bits, Xamarin - mobile specific bits.