At the top of the file there should be a short introduction and/ or overview that explains what the project is. This description should match descriptions added for package managers (Gemspec, package.json, etc.).
Show what the library, website, application, or game does as concisely as possible, developers should be able to figure out how your project solves their problem by looking at the code example. Make sure the API you are showing off is obvious, and that your code is short and concise.
A short description of the motivation behind the creation and maintenance of the project. This should explain why the project exists.
Describe and show with links to all resources used and attributions to all code used what other projects, libraries, apis, software, or third party resources this project requires to work as intended how you set up the functionality of this project with those resources. Be detailed with any downloads and installations and give an account of any problems you faced and how you solved them for future developers.
Provide code examples and explanations of how to get the project, how to run it as if it had never been run before (test this first by cloning the project as a fresh install into a new directory from git hub and describe the process of getting it running). Provide screen shots of the running code.
Depending on the size of the project, if it is small and simple enough the reference docs can be added to the README. For medium size to larger projects it is important to at least provide a link to where the API reference docs live.
Describe and show how to run the tests with code examples, include screenshots of your own testing efforts and the code working.
Let people know how they can dive into the project, include important links to things like issue trackers, irc, twitter accounts if applicable.
Describe any none issues or bugs with the project, how they were discovered and if possible describe how they come about. Provide at least one plan for overcoming these difficulties and multiple proposals for how to resolve them is better here for future contributers and your own continued efforts.
A short snippet describing the license (MIT, Apache, etc.)