The API is the top level API, of which you need at least one protocol driver (snarl, or HTTP), empire is our JPA interface, Sesame and Jena are obvious, and pack is the clustering support in Stardog 2.2.1 and later. Server side is just the server side components of all of the above. For embedded server applications, you need all of the jars in the server folder on the class path, and my favorite way of accomplishing that is using Gradle and the fileTree dependency declaration, but anything will do.
If you’d like to see more on client dependencies, the Maven dependencies are available via the “mavenInstall” script, and then you can use any build tool you like that can pull from your local maven repository (Maven, Ivy, Gradle, Leiningen, etc).
There’s also a listing of the top level dependencies and the client type in the docs: