A resource manifest is a class implementing the
IResourceManifestProvider interface though which you can declare your static resources (e.g. stylesheets and scripts) towards Orchard and give them a unique name to be able to easily use them in your templates. Choosing a unique name for each of resources is quite important to avoid name collisions, since static resources are rendered as shapes and thus can be overridden (see naming conventions).
This page and many others here are part of the open-source, portable package of Orchard 1.x goodies: the Orchard Dojo Library for Orchard 1.x. The Library is freely accessible by anyone; just go to its repository to download its full content (you can even download the textual content of the Library as a big concatenated document in HTML), file issues or fork it! You can even edit it online on Bitbucket, just click on the Edit link in the upper right corner!
Note that the Library also has an Orchard Core version, check it out here!