Adsorption of the alkali-, group-III, and 3$d$-transition-metal adatoms (Na, K, Al, In, V, Fe, Co, and Ni) on graphene was studied systematically by first-principles calculations. The bonding character and electron transfer between the metal adatoms and graphene were analyzed using the recently developed quasi-atomic minimal basis set orbitals (QUAMBOs) approach. The calculations showed that the interaction between alkali-metal adatoms and graphene is ionic and has minimal effects on the lattice and electronic states of the graphene layer, in agreement with previous calculations. For group-III metal adatom adsorptions, mixed covalent and ionic bonding is demonstrated. In comparison, 3$d$-transition-metal adsorption on graphene exhibits strong covalent bonding with graphene. The majority of the contributions to the covalent bonds are from strong hybridization between the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ and ${d}_{\mathit{yz}}$ orbitals of the 3$d$-transition-metal adatoms and ${p}_{z}$ orbitals of the carbon atoms. The strong covalent bonds cause large in-plane lattice distortions in the graphene layer. Charge redistributions upon adsorptions also induce significant electric dipole moments and affect the magnetic moments.