Well, fittype des not do the fitting itself. That would fall more into the regime of fit.
fittype merely allows you to specify the model. That model can be a variety of things, from a polynomial model, to a nonlinear model, to a spline interpolant, to a smoothing spline, etc. In the case of a spline interpolant, the fit is "exact", since it is an interpolant. In that case, the fitting algorithm is comepletely different. As well, in the case of a smoothing spline, the fit is now a smoothness penalized estimation. But in all the other cases, the fit is generally a simple least squares estimator. In the case of a some of the models one could choose, that fit will be what is called a linear least squares, where no starting values are necessary. A polynomial is a good example of this case.
But most models will fall into the nonlinear case. Again, this is just a simple nonlinear least squares. (You can choose between Levenberg/Marquardt or a trust region algorithm in the search.) The exeption is if you specify a robust fitting scheme to be used, but even that is probably just some form of iteratively reweighted scheme. So still the same optimization tools employed, but now with a scheme used to downweight large residual points, and then refit.
But essentially, if you are looking for the underlying mathematical method used, if the model is nonlinear, then a simple nonlinear least squares is the method. Nothing special under the hood in that respect.
