In basic terms, suppose you can measure some quantity which depends on several variables--say you can measure the hardness of concrete, which depends on the amount of various materials added to it. You test the hardness for a few material amounts to get some sample data. You want to somehow figure out the hardness at points in between your sample data. This is where surface fitting comes in--you want to fit a surface to your data (possibly in more than 3 dimensions, depending on your number of variables).

Lower dimensional interpolation (fitting) is pretty common. For instance, VBAhack's Cubic Spline Tutorial gives an example of single-variable interpolation. This post extends a single-variable interpolation to generate a surface fitting function. It does nothing to minimize error--that's another type of surface fitting you'd use if you had a theoretical model that predicted the relationships between your measured quantity and your variables.

This type of surface fitting would be useful if you wanted a "smooth" or "reasonable" fit when you only have a 1-dimensional interpolation handy. To keep the description reasonably short, I've used a fair amount of notation and jargon. Sorry; I included pseudocode.