This is a question I've been getting quite a bit lately. People have heard about it, but don't quite know what it is.

www.silverlight.net says it is:
Microsoft® Silverlight™ is a cross-browser, cross-platform plug-in for delivering the next generation of .NET based media experiences and rich interactive applications for the Web. Silverlight offers a flexible programming model that supports AJAX, VB, C#, Python, and Ruby, and integrates with existing Web applications. Silverlight supports fast, cost-effective delivery of high-quality video to all major browsers running on the Mac OS or Windows.

So what does it boil down to?

Microsoft Silverlight is something you install in your browser which allows you to view Silverlight content.

Silverlight content can be closely compared with Adobe Flash: rich design, animations, a statefull application that runs in the browser.

Silverlight is supposed to have better capabilities in integrating streaming video within your Silverlight application/website than Flash, but not having much experience with delivering video I can't tell from personal experience.

Silverlight content can be written using JavaScript or one of the many .NET languages.

Note 1: As of right now the plug-ins and tools for writing Silverlight content are still in beta.
Note 2: Silverlight was developed as WPF/E, which stands for Windows Presentation Foundation/Everywhere.
Note 3: Silverlight content can be viewed on Microsoft and Mac platforms. Novell has pledged a browser control based on Mono called Moonlight. This will bring Silverlight to the Linux platforms also.