Client Object Model (CO) is a new addition to SharePoint 2010.  You can develop applications in .NET, Silverlight or JavaScript because client Object Model provides assemblies for each of these technologies. Client Object Model consists of libraries that help us write code against .NET assemblies.  It is important to note that these libraries run outside the context of SharePoint.   The CO runs against a WCF service and addresses a lot of problems.  Previous versions of SharePoint provided web services but in order to use them, you needed knowledge of SOAP, XML and service endpoints that each service exposed.  There wasn’t a lot of consistency between the Server Object model and these web services.  Although, these web services are still supported in SharePoint 2010, new development against SharePoint 2010 should be done using the CO when remote connection to SharePoint Object Model is needed.  In addition, older services were not efficient because you had to use multiple services to perform basic operations.

Sharepoint 2010 has built in support for Silverlight. In order to use Silverlight, you have to use the new Cient Object Model. For an overview of the client object model, please see my post here