For a project, I have been asked to put a complete Facebook application inside the timeline. However, I cannot find any information about this in the Facebook reference. Does an
Probably you talking about Page Tab application for pages with Timeline.
Read the tutorial for Page Tab applications and documentation for Add to Page dialog.
For personal profiles applications in timeline is only section dedicated to actions and activities published by application to user.
I believe your client wants to use the new Add To TimeLine feature. This provides another introduction from user perspective.
As an example, this image shows Spotify, NetFlix and Nike app implementations of the new feature on the timeline:
You can implement a basic "add to timeline" functionality using the JavaScript SDK although API documentation is limited.
You can also create customized app contents and aggregate contents ("movies watched this month on Netflix") for the Timeline by integrating your app with the Open Graph. Facebook has this introductory tutorial.