Well at first attempt it will sound like, what ? ASP.NET application is processed on server side, how will you get the output?. Well remember its not always that you want to run a visual application. it can be a background application like some utility. For eg: After uploading a file you want to zip it and store it on your disk. Infact any console output emitted by your application can be read and displayed back into webbrowser
You can use your System.Diagnostics namespace to interact with system processes. Here is a small tip.
Note that the ASP.NET worker process needs to have permissions to access the external application. The most simple way to ensure this is to place it under the bin folder.