Ok, so, this isn't a straight forward question. Models like the very general supply and demand model, or a Solowian growth model, or a Heckscher-Ohlin model trade model etc. etc. are big M models which aim to explain how/why something happens or maybe even will happen. The actual effect of minium wage policies is an empirical question, but empirics is difficult because you're trying to measure the real world which is influenced by a ton of different things, so you need empirical models to do this. You can't just observe that a minimum wage policy happened and then unemployment changed by X some time later and that's that.
In
this (pdf file) relatively new working paper Dube et.al. use an empirical model to analyze the effect to 138 minimum wage changes over a 35 year period, where they use bunching (they look at changes in employment directly below and above the minimum wage treshold). This isn't really what most people have in mind when talking about models, though.
There are several different models looking at the labor market specifically, but the easiest is to just think of it in the same supply and demand framework I described above but where employers have some degree of market power. So instead of that graph with two lines you see when you google "supply and demand" you get something like this:
With perfect competition the market clears where supply intersect with demand, so at E1 employment at a wage of W1. If employers have market power then you get that third line (because employers can lower the market wage by restricting how many they employ, with full competition they can't), so market wage will be the lower W3 at the lower employment level E2. If a minimum wage is set at a level higher than W3 but not exceeding W1 then wages will rise and employment increase. Above W1 you'll start getting the same effects you see if you google "supply and demand price floor". This is basically what Dube et.al. talks about
here (a monopsony is the same thing as a monopoly, but from the demand side instead of the supply side).
So this basically non-answer is that Dube is looking at the labor market using the classic supply and demand model/framework described in Econ 101, but where employers have some degree of market power. He's using a wide variety of empirical models to isolate the effect of the minimum wage policies on wages and employment, and he's then using those empirical findings to predict the effect of future minimum wage policies.