Addictive post by Andrew Gelman on That Modeling Feeling, the rest
It goes like this: there's something you want to estimate and you have some data. Maybe, to take my favorite recent example, you want to break down support for school vouchers by religion, ethnicity, income, and state (or maybe you'd like to break it down even further, but you have to start somewhere).
Or maybe you want to estimate the difference between how rich and poor people vote, by state, over several decades--but you're lazy and all you want to work with are the National Election Studies, which only have a couple thousand respondents, at most, in any year, and don't even cover all the states.
Or maybe you want to estimate the concentration of cat allergen in a bunch of dust samples, while simultaneously estimating the calibration curve needed to get numerical estimates, all in the presence of contamination that screws up your calibration.