With the synthesis of a single atomic plane of graphite, namely graphene honeycomb structure, active research has been focused on the massless Dirac fermion behavior and related artifacts of the electronic bands crossing the linearly at the Fermi level. This thesis presents a theoretical study on the electronic and magnetic properties of graphene based structures, and phononic energy dissipation. First, functionalization of these structures by 3d-transition metal (TM) atoms is investigated. The binding energies, electronic and magnetic properties have been investigated for the cases where TM-atoms adsorbed to a single side and double sides of graphene. It is found that 3d-TM atoms can be adsorbed on graphene with binding energies ranging between 0.10 to 1.95 eV depending on their species and coverage density. Upon TM-atom adsorption graphene becomes a magnetic metal. TM-atoms can also be adsorbed to graphene nanoribbons with armchair edge shapes (AGNRs). Binding of TM-atoms to the edge hexagons of AGNR yield the minimum energy state for all TM-atom species examined in this work and in all ribbon widths under consideration. Dependingon the ribbon width and adsorbed TM-atom species, AGNR, a non-magnetic semiconductor, can either be a metal or a semiconductor with ferromagnetic or anti-ferromagnetic spin alignment. Interestingly, Fe or Ti adsorption makes certain AGNRs half-metallic with a 100% spin polarization at the Fermi level. These results indicate that the properties of graphene and graphene nanoribbons can be strongly modified through the adsorption of 3d TM atoms. Second, repeated heterostructures of zigzag graphene nanoribbons of different widths are shown to form multiple quantum well structures. Edge states of specific spin directions can be confined in these wells. The electronic and magnetic state of the ribbon can be modulated in real space. In specific geometries, the absence of reflection symmetry causes the magnetic ground state of whole heterostructure to change from antiferromagnetic to ferrimagnetic. These quantum structures of different geometries provide novel features for spintronic applications. Third, as a possible device application, a resonant tunnelling double barrier structure formed from a finite segment of armchair graphene nanoribbon with varying widths has been proposed based on first-principles transport calculations. Highest occupied and lowest unoccupied states are confined in the wider region, whereas the narrow regions act as tunnelling barriers. These confined states are identified through the energy level diagram and isosurface charge density plots which give rise to sharp peaks originating from resonant tunnelling effect. Finally, we studied dynamics of dissipation of local vibrations to the surrounding substrate. A model system consisting of an excited nano-particle which is weakly coupled with a substrate is considered. Using three different methods, the dynamics of energy dissipation for different types of coupling between the nano-particle and the substrate is studied, where different types of dimensionality and phonon densities of states were also considered for the substrate. Results of this theoretical analysis are verified by a realistic study. To this end the phonon modes and interaction parameters involved in the energy dissipation from an excited benzene molecule to the graphene are calculated performing first-principles calculations.