In recent years, there has been an increasing interest in probing the properties of two-dimensional materials, starting with graphene. This one-atom-thick sheet has important features. Thus, we employ an accurate quantum many-body formalism, the coupled-cluster method, to study the electronic correlation effects on the quantum state of two-dimensional materials described by Hubbard model. In particular, we examine the ground and excited states of the model.To get a quantitative description of the quantum properties of the Hubbard model, we calculate the ground and excited state energies in addition to the staggered magnetisation which is the ordered parameter considered in this work. We report results for the 1D chain, honeycomb lattice, and in a few occasions, also for square lattice.We first of all investigate number of approximations within the normal version of the coupled-cluster method. With hoping terms only, one-body correlations are significant and it is capable of producing the exact results of the 1D chain. For the square and honeycomb lattices, the results are identical to the solution of mean-field theory in that limit. At strong coupling, only nearest-neighbour spin fluctuations survive, and the results converge as expected to those of the Heisenberg model. However, the approximation fails to predict the existence of a symmetric state in the 1D chain at non-zero coupling and does not show a phase transition between a semi-metal and antiferromagnetic phases in the honeycomb lattice. The single charge excitation converges, at zero-coupling, to the dispersion relation of the tight-binding model and there is no effect of the two-body correlations in that limit. One approximation, which is based on a model state that embodies long-range spin fluctuations, succeeds to produce a symmetric state for the 1D chain. However, it fails to show a phase transition in the honeycomb lattice.Next, we apply the extended version of the coupled-cluster method. One approximation in particular shows results close to mean-field solution at intermediate coupling and converges to the Heisenberg solution at strong coupling.