This explanation is aimed at consumers rather than developers and deliberately glosses over some things.  The result is probably over simplified for developers.
Silverlight is a technology which allows you to run an executable ( a programme )  in a browser.  Usually a browser "just" renders html as a sort of document.  Javascript can allow such a document to do things which appear reasonably dynamic.  Buttons which "do" things etc.
HTML5 has added support for things like showing videos.
The notable difference is that none of this is about running compiled code. 

Silverlight comes in two pieces which allow you to run compiled code, inside a web page.
  1. Browser Plugin
  2. Silverlight Framework
The PlugIn allows a Silverlight based application to be loaded as an object into your browser.
The Framework is a set of support libraries which supply common functionality a Silverlight application will use.
The thing to run is delivered as a .xap file - you can think of this as being a sort of exe.

That xap file can do all sorts of things, depending on what the developer who wrote it had in mind.
For many consumers that means some sort of video player but it can be other things.

Not Full .Net

Silverlight does not depend on the full .Net framework.  Some people experiencing problems have tried installing developer tools including the full visual studio and .Net framework in order to resolve issues.  This is not going to help.
The Silverlight Framework is a totally different thing to .Net. Similar, but different.

See Also

SIlverlight Resources on the Technet Wiki

Other References

About Silverlight
Silverlight - Wikipedia