.NET Core is a general purpose development platform maintained by Microsoft and the .NET community on GitHub. It is cross-platform, supporting Windows, macOS and Linux, and can be used in device, cloud, and embedded/IoT scenarios.
The following characteristics best define .NET Core:
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Flexible deployment: Can be included in your app or installed side-by-side user- or machine-wide.
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Cross-platform: Runs on Windows, macOS and Linux; can be ported to other OSes. The supported Operating Systems (OS), CPUs and application scenarios will grow over time, provided by Microsoft, other companies, and individuals.
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Command-line tools: All product scenarios can be exercised at the command-line.
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Compatible: .NET Core is compatible with .NET Framework, Xamarin and Mono, via the .NET Standard Library.
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Open source: The .NET Core platform is open source, using MIT an Apache 2 licenses. Documentation is licensed under CC-BY. .NET Core is a .NET Foundation project.
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Supported by Microsoft: .NET Core is supported by Microsoft, per .NET Core Support
More on the Official .NET Core Documentation from Microsoft
Basically, Microsoft built a version of .NET to allow developers to write cross-platform and cloud-optimized applications.
First, .NET is cross-platform. It runs on Windows, macOS and Linux, which allows the developer to share and run the exact same code between machines running different operating systems, with no changes in code and minimum or no changes in the development process. (Watch out for OS specific APIs!)
But the true innovation in .NET came with the modularization in design and architecture. The compiler (Roslyn) and the runtime (CoreCLR) are separate components that allow you to use different implementations (or even write your own).
Every library comes as a NuGet package, so when you start a new project, you don’t have any libraries, but a project file (currently a .json, in the future a .csproj to maintain consistency with all .NET projects) and a Program.cs. As you develop your app, you add libraries as you need them, allowing you to minimize the size of your application.
.NET Core also allows you to have multiple versions installed at the same time without having apps / parts of the OS breaking when you update or install a newer version of the framework, and will even allow you to ship the framework / parts of the framework with the application (since the footprint of the framework si small enough).