It's possible to implement a web server with Windows 10 Core for IoT on the Raspberry Pi 2. If you access your device on http://devicename/, you will actually see a website that is already provided by the system.
To implement your own web server on Windows IoT, there are a bunch of possibilities. For Python or Node.js, there are already samples available on the official GitHub project:
As you mentioned ASP.NET: I didn't stumble across any sample yet, but you should be able to do that, either with a classic self-hosted ASP.NET 4 application or you could give it a try with ASP.NET 5 (aka "ASP.NET vNext").
In both cases, you should create a Windows Universal App first, which will host the web server. Now, you can follow the same approach as described in several tutorials out there that demonstrate how to implement a self-hosted ASP.NET server, e.g. using a console application. Instead of a console application you are using the Windows Universal App here, but that basically doesn't make a difference.
ASP.NET 4 sample :
ASP.NET 5 sample:
Hope that helps!