Graphene is an allotrope of carbon. Recently , graphene has emerged as a fascinating system for fundamental studies in condensed matter physics. The discovery of graphene and its remarkable electronic and magnetic properties initiated great research interest in this material. The Nobel Prize in Physics for 2010 was awarded to Andre Geim and Konstantin Novoselov at the University of Manchester for groundbreaking experiments regarding the two dimensional material graphene. Graphene is a one atom thick layer of graphite, where low-energy electronic states are described by the massless Dirac fermion, so that the unique features of graphene electronic properties arising from its gapless, massless, chiral Dirac spectrum are highlighted. Furthermore Graphene nanoribbons are candidate materials for future applications in nanoelectronics and molecular devices due to their semiconducting properties. Finite graphite systems having a zigzag edge exhibit a special edge state. The corresponding energy bands are almost flat at the Fermi level and thereby give a sharp peak in the density of states. The charge density in the edge state is strongly localized on the zigzag edge sites. No such localized state appears in graphite system having an armchair edge. Moreover, the quality of the draphene derivatives are so good that ballistic traort and quantum Hall effects (QHE) have been observed. Among graphene derivatives, graphene nanoribbons constitute a fascinating object due to a rich variety of band gaps, from metals to widegap semiconductors. In particular, the half filled zero energy states emerge in all zigzag nanoribbons and hence they are metallic. Another basic element of graphene derivatives is a graphene nanodisk. It is a nanometer scale disk like material which has a closed edge. It is also referred to as nanoisland, nanoflake, nanofragment or graphene quantum dot. Nanoribbons and nanodisks correspond to quantum wires and quantum dots, respectively. They are candidates of future carbon-based nanoelectronics and spintronics alternative to silicon devices. In this project, the electronic quantum traort properties of graphene nanoribbons are studied. Although these systems share the similar graphene electronic structure, confinement effects are playing a crucial role. The lateral confinement of charge carriers could create an energy gap near the charge neutrality point, depending on the width of the ribbon.Then the conductance of metallic graphene nanoribbons with single defects and weak disorder at their edges is investigated in a tightbinding model. We find that a single edge defect will induce quasilocalized states and consequently cause zero conductance dips. The center energies and breadths of such dips are strongly dependent on the geometry of graphene nanoribbons. Armchair graphene nanoribbons are more sensitive to a vacancy than zigzag graphene nanoribbons, but are less sensitive to a weak scatter. More importantly, we find that with weak disorder that is modeled with the Anderson model, zigzag graphene nanoribbons will change from metallic to semiconducting due to Anderson localization. However, weak disorder only slightly affects the conductance of armchair graphene nanoribbons near the Fermi energy. Keywords: Graphene nanoribbon, Quantum traort, Coherent traort, Disordered systems, Weak scatterer, Vacancy, Landauer formalism.