If you want to develop facebook applications, you should add the "Developer" app to your applications in facebook, or i suggest that you create a dev account for this.
http://www.facebook.com/developers/?fa=1
Now the second step would be having a look around the available APIs, don't get intimated by the mess and disarray of facebook APIs, they are a mess but with enough effort you can get to where you want.
Mainly there are three APIs that you should know about:
Graph API - A RESTful api for accessing user data
FQL API - Access data through sql-like queries
Javascript API - for setting up quick authorization mechanism, and lots of other goodies
The authentication mechanism they use is OAuth2.0 which is great and simplified, you don't have even to be bothered to know that is OAuth2.0 because the model is really straightforward(gosh unlike the linkedin OAuth1.0 mess)
Facebook provides an official library for php, you might want to have a look at it sometimes.
You can consume facebook API in two formats: JSON or XML
The de-facto choice is JSON because it is has a really smaller foot-print(size) thus better performance.
Consuming JSON in php is really straightforward because it is a dynamic language, .net on the other hand is somewhat more irritating, you have multiple choices to parse your JSON in .net, i suggest that you have a look at James Newton-King Json library, and at the datacontract serializer. I also did not have a look at the possibility of handling JSON object via dynamic objects in .net 4.0, you might also want to research this.
Edit: a simple detail that i have forgotten, if you wish to use the javascript api which you probably will, you can't test on a local dev server, you should deploy to somewhere like live(test server), yup that is really da huge pain in da b**** when developing for facebook
Now for your scenario, if you only want the user to send invitations, the javascript api might be sufficient so you don't get to get in this whole mess/just half of it :p
http://developers.facebook.com/docs/reference/dialogs/requests/
but remember you need to create a facebook app, add the integration snippet in your master page and test this on a live server somehwere