Summary: Flex is to Adobe as Silverlight is to Microsoft
Silverlight is often referred to as Microsoft's version of Flash, however it is more correct to say it is Microsoft's version of Flex.
You could say the following…
Silverlight 1.0 = Flash
Silverlight 2.0 = Flex
[UPDATE: Silverlight 1.1 has been re-branded as Silverlight 2.0]
Here's a very basic comparison of Flex and Silverlight:
Flex is an Adobe product which allows you to build Rich Internet Applications* (RIA)
It is basically a developer tool which has a framework of functions that allows you to build applications that run in a browser (any browser on any platform).
Flex can use the Adobe Integrated Runtime (AIR)** to run those very same web apps on the desktop as desktop apps.
Flex is currently building in .Net framework support.
Flex is currently in Beta for version 3.
In comparison, Silverlight is a Microsoft technology that allows you to build RIAs.
Silverlight apps are built using Visual Studio, Blend and other tools.
Silverlight 1.0 (released in October 2007) focused on providing Rich Media support.
Silverlight 1.1 (currently in Alpha) is focused on the programmability behind the apps.
Via a Silverlight plug-in, the apps run in any browser on any platform (there are some limits though).
Silverlight does not provide desktop app functionality. WPF is the Microsoft solution for rich media desktop applications.
* Some people contest that RIA stands for Rich Interactive Applications
** AIR used to be referred to as Apollo (it's previous codename)