This work presents a systematic review of the feature-rich essential properties in graphene-related systems using the first-principles method. The geometric and electronic properties are greatly diversified by the number of layers, the stacking configurations, the sliding-created configuration transformation, the rippled structures, and the distinct adatom adsorptions. The top-site adsorptions can induce the significantly buckled structures, especially for hydrogen and fluorine adatoms. The electronic structures consist of the carbon-, adatom- and (carbon, adatom)-dominated energy bands. There exist the linear, parabolic, partially flat, sombrero-shaped and oscillatory band, accompanied with various kinds of critical points. The semi-metallic or semiconducting behaviors of graphene systems are dramatically changed by the multi- or single-orbital chemical bondings between carbons and adatoms. Graphene oxides and hydrogenated graphenes possess the tunable energy gaps. Fluorinated graphenes might be semiconductors or hole-doped metals, while other halogenated systems belong to the latter. Alkali- and Al-doped graphenes exhibit the high-density free electrons in the preserved Dirac cones. The ferromagnetic spin configuration is revealed in hydrogenated and halogenated graphenes under certain distributions. Specifically, Bi nano-structures are formed by the interactions between monolayer graphene and buffer layer. Structure and adatom-enriched essential properties are compared with the measured results, and potential applications are also discussed.