The procedure is essentially an evaluation of a millions-of-dimensions-large integral via Markov-chain Monte Carlo. Each step in the Markov chain requires inverting extremely large ill-conditioned matrices, which we can make better-conditioned by moving away from physical parameters (and then trying to extrapolate). People are glad to have only hundreds or thousands of independent samples in an ensemble. You need ensembles for multiple lattice spacings and spatial volumes in order to ensure you can control your discretization and finite-volume artifacts when you extrapolate to the continuum and to infinite volume.
What is generated in the Markov chain is essentially the configurations of the gluons and quark loops. With these configurations in hand we can measure physical observables on each.
Observables with baryons have an exponentially bad signal-to-noise problem in the temporal direction (time on the grid, not Monte Carlo time) and you either need an exponentially large computer, an exponentially long time, or an exponentially wealthy funding agency. There are attempts to circumvent this issue, we'll see how successful they prove...