A .NET Core Class Library is built upon the .NET Standard. If you want to implement a library that is portable to the .NET Framework, .NET Core and Xamarin, choose a .NET Standard Library
.NET Core will ultimately implement .NET Standard 2 (as will Xamarin and .NET Framework)
.NET Core, Xamarin and .NET Framework can, therefore, be identified as flavours of .NET Standard
To future-proof your applications for code sharing and reuse, you would rather implement .NET Standard libraries.
Microsoft also recommends that you use .NET Standard instead of Portable Class Libraries.
To quote MSDN as an authoritative source, .NET Standard is intended to be One Library to Rule Them All. As pictures are worth a thousand words, the following will make things very clear:
1. Your current application scenario (fragmented)
Like most of us, you are probably in the situation below: (.NET Framework, Xamarin and now .NET Core flavoured applications)
2. What the .NET Standard Library will enable for you (cross-framework compatibility)
Implementing a .NET Standard Library allows code sharing across all these different flavours:
For the impatient:
For a table to help understand what the highest version of .NET Standard that you can target, based on which .NET platforms you intend to run on, head over here.
Sources: MSDN: Introducing .NET Standard