Let's start at the start. Before ASP Core there was ASP (Well there were many other things but that's not the point) ASP is old, it runs only on Windows but it's mature and isn't really going anywhere soon so don't panic unless your reading this in the future and this is the only reference to ASP.
ASP needed the .net framework to be installed on the computer you wanted to run it on which meant updates and reboots.
Microsoft go OSS mad
Unless you have been hiding Microsoft are going open source mad at the moment. .NET is going open source of course since it's going open source it means a pretty solid rewrite of the code base to refactor it(we all know they mean to take out those comments). As they move to open source with .NET they are also making it cross platform. So they need a new web framework for .NET Core
Enter ASP Core
ASP Core is the next version of ASP (originally known as ASP 5 or ASP vNext) but at least someone showed sense to rename it and change it because it is not ASP 5. It's very different to ASP.
ASP Core a new open-source and cross-platform framework for building modern cloud-based Web applications using .NET Core. It was built it from the ground up to provide an optimized development framework for apps that are either deployed to the cloud or run on-premises. It consists of modular components with minimal overhead, so you retain flexibility while constructing your solutions. You can develop and run your ASP Core applications cross-platform on Windows, Mac and Linux.
ASP Core is open source and up on GitHub.
As soon as RC2 is locked in I'll do another post on the difference between ASP 4 and ASP Core
There are a few ways to get started. At the time of this post ASP Core is on RC1
The best place to start would be from DotNet on GitHub This is a step by step of the stable version.
If you want to live dangerous then try the latest version from the dotnet cli Scroll down to the installers get the ".NET Core SDK Installer" Check out this ASP Community Standup Damian Edwards does a great walk through of the process to get the latest bits up and running. Also the ASP Community Standup is a really great resource for all things ASP.