I have the source code of a web appliaction which has .aspx,.asax,.ascx,.cs,.js and other related files.Can anyone tell me how to run this web applicatin?

You have an asp.net application. You will need to create a directory on your filesystem and point an IIS virtual directory there. From there, you can run the site from your local installation of IIS.

Optionally, you can load the files you have into your Visual Studio 2008 or Visual Web Developer Express 2008 and run/debug the source code from there.

....but for starters you need the correct .NET Framework installed on the machine running the IIS.
Also you may need to setup an IIS application to run the web site in question.

As regards VWD: I myself have no experience with the 2008 versions, but trying to open a 2003 (.Net 1.1) project in a 2005 version will result in a mandatory project conversion which almost always goes bad. So...I you can, install and use the apropriate version of VWD for the project. A bad conversion will require you to hand-edit the faulty code.