In simple terms when you only want to refresh only part of a page rather than posting back the whole page to
the server in that case AJAX Provides many ways and many controls to do it.It is Asynchronous Communication.
see this link to know more about AJAX
Whereas Silverlight is a plugin powered by the .Net Framework for creating interactive user experiences for developing
web, desktop and mobile applications.
You might have heard about Flash, Silverlight is similar to that(Microsoft Product)
Silverlight has got many more features compared to Flash which many designers and developers always wanted to have
see below link
In the above link you will find numerous number of tutorials, how do i videos, follow then to learn more