They were all talking about differential equations, just some of you don't know it. Global circulation models are a collection of coupled atmosphere, ocean, etc models. Each of these models contain a core set of differential equations, which are either discretized to be integrated forward in time in physical space, or decomposed into spectral space, which has certain benefits for non-linear terms in the Navier-Stokes equation. There are a number of parameterizations to handle sub grid-scale processes so their effects taken into account at the resolved grid scale*. In essence you have a bunch of differential equations and a closure to give yourself a closed system for each component of the GCM, which you then use to force other components, and you integrate it all forward in time.
And the gp was right about observations. If you recall your ODE/PDE class, you'll be interested to know this is a boundary-value problem and you need to specify initial and boundary conditions. Initial conditions are your observations, or whatever your assumptions about the current state are. Often the GCM models are initialized in the year 1800 or 1900, giving them 100+ years of simulation time to equilibrate and match known observations before they are really forecasting the future. As for boundary conditions, the model is global, so the boundaries wrap around and you dont need to worry about them.
* An example of this is convection. When moist air rises and condensation occurs (to form cloud drops, rain, ice, etc), energy is released into the surrounding system (enthalpy of vaporization, deposition, fusion, etc). This translates into warming of the surrounding air, and helps drive convection and represents a transport of warming from the surface to the middle and upper atmosphere. The condensation process happens on a much smaller scale than a GCM can resolve, so the equations being integrated cannot represent this process. The process does however have an effect on temperature at the resolved scale. To handle this, parameterizations are employed that make certain assumptions about these processes and then make adjustments to the resolved scale. It would be better to just resolve these effects directly, but when you try to work at the molecular scale globally, realtime moves faster than the model does.