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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, in effect, a slow backward model can be devised from a fully-formed forward model, by searching the parameter space; clever, efficient searches can sometimes be devised, making this less slow. A backward model that is efficiently calculable is ideal in analysis involving large numbers of cases of observed data, e.g., from photometric surveys of many stars.