Box modeling is where you start with a shape that is closed on all sides (such as a cube, thus the name "box") and extend the faces. This method is often touted as being good for making non-organic things like buildings, that have naturally sharp edges.

Polygon modeling is kind of the same thing, but focuses on models where the edges are not all connected. This is more like starting with a flat sheet of paper instead of a cube and going from there.

Spline modeling is... well, first of all, a spline is a kind of curved line, that's defined by control points. You can make very elegant curves with a limited amount of input. These lines can be extended out into surfaces. Splines were invented originally to help model cars in the computer, as the french curve was breathing its last, gasping breath. Many people like to work with non-uniform rational B-splines (most people just say NURBS, I'm kind of a geek and love my tech-speak).

There are many combinations of these kinds of modeling, though, and they're just general terms. For example, I just learned in Carrara how to make a spline, match it with a separate polygon, and make an extruded curved box. Kind of a combo of all three modes.

Also, before someone tells you otherwise, there is no "best" way. What you really want to do is work with a modeler and learn how it works, without worrying too much about one style of modeling over another. As with all things, modelers and modeling methods have their rabid, die-hard fans who won't hear a word against their particular toys.