Well, this topic takes up six pages in Ralph Mayer's "The Artist's Handbook" (5th edition) but I'll take a stab at it:
Sables and their imitations, soft fine bristles, are for delicate work where you want to minimize the brush marks. They are very good for glazing techniques. Bristle brushes or hogs hair brushes have stiffer thicker bristles and leave more texture in the paint marks. They can also handle heavier bodied paint for more impasto styles, where the sables are better for more watery paint and especially water colors.
Rounds and pointed rounds are good for more fluid paints and a simpler line shape, large ones are great for covering a lot of area with color. Flats are considered the most versatile since you can get a good variation in line from them. Filberts are like a rounded flat, kind of oval shaped, so fall in between the two, I find them good for hitting very specific spots. Brights are "good for buttery paints," so putting down thicker layers, they are a little stubby so are more likely to pick up lower layers when applying wet on wet if you work them too much.
Specialty brushes like chisels and sign painter brushes (very long flats) are all about the specific lines they produce. Fan brushes, another specialty brush, are good for very light applications and for softening existing brush marks (one of my favorite kinds of brush FWIW, think blur tool in photoshop). Water colors also have special brushes like mops which are good for putting down washes as opposed to the points that are more for detail work.
Hope that helps. To me it comes down to how much texture the brush puts in the paint (bristles), and the feel of the line it makes (length and shape).