Creating an Application in Facebook
If you want to create or to make your own application in Facebook its very easy, and there are some pages inside facebook and some applications that will help you in achieving that. In this page you can create your own application of phrases: Creator of Phrases (to create applications of phrases) you select the one that you want and you follow the instructions, in just a short time you will have your application working.
Also you have other pages that allow you to create more applications for Facebook in simple steps, with this one you will be able to create: applications of interviews, with Social Interview, with it you will be able to start your own interview on facebook.

There are many other pages in facebook that will allow you to create applications of of phrases, applications with feeds, application of X of the day, tests in Facebook, applications of gifts, applications of photos, surveys in Facebook…

If you are pretty serious on creating apps for facebook you should check the site for the stanford course for developing facebook applications. You can find more info in their facebook page: Stanford Course, Creating Apps for Facebook
Here is a link to the current list of apps the class has developed:
http://tinyurl.com/2ku777

Also you can Create a Facebook App with Google App Engine, basically google app engine is a service that allow developers to create and host application with different purposes. In this page you will find useful information on hosting a Facebook app on Google AppEngine.

For serious developers with knowledge of coding and programmation theres is the facebook developer Wiki: http://wiki.developers.facebook.com/index.php/Creating_a_Platform_Application

HOW TO ADD MY APPLICATION TO THE FACEBOOK DIRECTORY?
Once we already have our application we will be able to add it to the directory of Facebook, in this page they give all the details for adding your new app to the facebook evergrowing directory of applications.