Core is the main module for ProjectRed. Overall, it doesn't add anything useful or exciting to the game. It serves as a library of utilities, along with CodeChickenLib, that are required for anything else to work. It also manages the communications between the different modules in PR.
The module adds a number of common crafting ingredients as well as a couple of basic tools to the mod. These are all shared between other modules as common ingredients or tools that can affect any module with parts relevant to their function. Interestingly, though core provides gems and metal ingots, Exploration is the module that actually adds the ores and blocks for them.