In this thesis we present calculations in lattice quantum chromodynamics. We begin with an introduction to the general non-perturbative methods that are used. This includes the chirally improved Dirac operator and the quenched approximation which is applied for the calculations. After a short look at the multiplet structure of baryons we study positive and negative parity nucleons. Our analysis is based on a set of three operators with the nucleon quantum numbers. We also explain the variational method used to separate the ground state and the excited states. With this method it is furthermore possible to determine the mixing coefficients for the optimal nucleon operators in terms of our initial set of operators. In the presented results we can clearly identify the negative parity resonances N(1535) and N(1650) and find that their masses agree well with experimental data. On the other hand, the mass of the observed positive parity excited state is too high to be interpreted as the Roper resonance. Our results for the mixing coefficients indicate that chiral symmetry is important to investigate the negative parity spectrum of the nucleon.