A forward model is a model that takes parameters and produces observable values. Thus the model plus a set of parameters constitutes a prediction. A backward model (or inverse model) is a model that works in the reverse direction: given values observed, produces parameters, which could be characteristics not subject to observation.
For example, a backward model might be a means of calculating the abundances of constituents of an extra-solar planet's atmosphere from transmission spectroscopy data, whereas a forward model would take assumptions of the abundances and calculate what the transmission spectroscopy data would be.
Backward models often can be devised using Markov chain Monte Carlo methods. Also, a slow backward model can be devised from a forward model through searching the parameter space. A backward model that is efficiently calculable can, in turn, be used in analysis involving many cases of observed data, e.g., from photometric surveys of many stars.