XAML vs WPF vs Store App vs Phone App vs UWA
I would highly recommend referring to this discussion thread in stopbyte, it has far more details and explanation of the concepts behind WPF, UWP And WinRT.
As for WinForms, from my experience it's history now. Have been almost replaced by WPF (though it still has mainstream support as mention by @Jazimov). I'v noticed that Microsoft has stopped improving Windows Forms and probably has stopped maintaining it as well.
When .NET was released in ~2000, you could use WinForms to make desktop applications for Windows. It provides access to 'native' controls (i.e. A Button made in WinForms actually is a Windows Button control from user32.dll) by wrapping the Windows API in managed code. These can run in any version of Windows which has the appropriate .NET framework version installed, including upto Windows 10.
In 2006, WPF was released which is an alternative to WinForms. WPF uses XAML, which is a language based on XML, to declare the user interface elements. In a simple WPF app, the .xaml file describes the GUI and the code-behind file describes the logic. DirectX is used to draw whatever is described in the .xaml into the window.
Like Winforms, programs made using WPF can run in any version of Windows which has the appropriate .NET framework version installed, including upto Windows 10. Note that using XAML is completely optional. Whatever user interface elements you describe using XAML can also be made using your .NET language C# or VisualBasic.NET. But using XAML is usually simpler. See To XAML or not to XAML (MSDN Blogs).
From XAML (Wikipedia) :
As XAML is simply based on XML, developers and designers are able to share and edit content freely amongst themselves without requiring compilation.
From ~2012 along with Windows 8, a new kind of app called metro/modern/windows-8/windows-store-apps can be made. These apps are similar to WPF in that they use XAML for describing the GUI. These apps can run only in Windows 8, 8.1 and Windows 10. Windows 8.1 added some improvements and features but remained mostly the same. The same technology was used to make apps for Windows Phone.
With the release of Windows 10, Microsoft made UWP (Universal Windows Platform) through which you can use the same code base to target differnt kinds of devices (desktop/mobile/xbox/IoT/holographic). These apps made for the UWP are similar to the earlier Windows 8/8.1 apps. Like before, XAML is used to describe the GUI.
Only for the Windows Store apps, there is no backward compatibility. The timeline goes Windows 8 -> Windows 8.1 -> Windows 10/UWP. An app made in UWP can only run in Windows 10 and an app made in Windows 8.1 can not run in Windows 8.