First, there is not a simple application form which one can fill out and submit to the patent office.
A patent application comprises several components; fees, a Petition setting forth the purpose of your application including who the inventor(s) are, and a technical paper describing the invention.
This technical portion is what is usually called the patent application and includes the following major portions: Field of the Invention, Background of the Invention, Summary of the Invention, Brief Description of the Drawings, Detailed Description of the Preferred Embodiments, the Claims, and the Drawings.
In the patent application, you must describe the prior technology or art, the invention and its solution over the prior art. You must also clearly disclose the method or best mode of making or carrying out the invention. This disclosure must be enabling, allowing another of the same skill to carrying out or make the described invention.
Another important part of the application is the claims which set forth exactly what you regard as being your invention and what you claim as being your exclusive property or privilege to make use or sell. These are located at the end of the application. Each claim must be written in a single sentence format.