What is a breadboard (retrieval) algorithm in the context of geophysical satellite retrievals/remote sensing?
When determining the retrieval algorithm for a spectrometer that will be built, it is first done on highly mathematical software platforms.  The software allows to the best curve approximation fitting equations to be derived (across all detected wavelengths) that will work for the proposed optics.  This is important because the diffracted light that is detected will interfere with itself very uniquely, depending on the geometry of your optical setup.   The data from the spectrometer's detector needs to be interpreted with a good curve fitting function so that you can get good wavelength resolution from your instrument.  

When building a spectrometer for operational use on a satellite, it is ideal to be able to process the retrieval algorithm using hardware, rather than software.  The hardware processor allows real time function that isn't dependent on an operating system that requires lot's of computational overhead.  So, you build a breadboard that you can connect to your spectrometer so that your retrieval algorithm is processed using integrated circuits that can be programmed.  You essentially hard-code your algorithm into hardware.

Search for "breadboard spectrometer" in Google and you will find many examples.  

Now, to answer your question.  A breadboard algorithm is developed using software that is used to help engineer the actual breadboards to be built in the future.  The software allows you to predict the response of your proposed circuitry to the spectrometer detector without having to actually build the breadboard yet.  Once your breadboard algorithm is setup properly (confirmed by testing the software version of your needed hardware) you can go about building the actual breadboard spectrometer model.

Once a breadboard spectrometer model is built and passes quality control checks, you can evaluate the spectrometer performance and see if its worth optimizing and fabricating a standalone instrument payload that can be deployed in the field (or in space).