Quantum Chromo-Dynamics (QCD) is believed to be the theory of the strong interaction of nature. We investigate some of the properties of QCD using the lattice formulation. In particular, we use non-relativistic QCD; NRQCD, to simulate heavy quark bound states. We perform these simulations on gauge configurations (vacuum snapshots) provided by the MILC collaboration [1]. These gauge configurations arc the first to include a realistic description of the full complexity of the QCD vacuum by incorporating the effects of light dynamical quarks. In doing this, they remove a major source of uncertainty from this type of calculation and allow us to make precise and relevant determinations. We simulate the Bc meson by calculating the b¯ quark propagator in NRQCD and combining it with a c quark propagator generated by our collaborators using a different quark formulation. We perform a range of checks to analyse the possible systematic corrections to our calculation then we proceed to extract the first precise prediction of the mass of the Bc meson from dynamical lattice QCD. The spectra of the b¯b and c¯c systems are simulated using the coarsest ensemble of configurations produced by MILC. We use the calculated splittings in these systems to assess the viability of the ensemble for further determinations. We then examine the leptonic width (width for decay into two leptons) of the Upsilon and J/psi vector states. These calculations extend earlier work on the b¯b system using the coarse and fine MILC ensembles, and they look for the first time at the c¯c system using NRQCD to describe c quarks. Additionally, we use NRQCD to describe both of the heavy quarks in the Bc system, allowing us to give a hadron mass prediction from NRQCD alone. Finally, we examine the Bs, B, Ds, and D systems by calculating NRQCD heavy quark propagators and combining them with light quark propagators provided by our collaborators. Again our calculations in the b quark systems add to results previously obtained on the MILC coarse and fine ensembles, while our c quark calculations are the first of their kind using an NRQCD heavy quark. Initially we look at the spectroscopy of these mesons, then we analyse their decay constants, which, for the Bs/B systems in particular, are experimentally very relevant at this time.