Saber grip: handle held across the palm, as opposed to straight up and down, with thumb along either the spine or side of blade.
Hammer grip/natural grip: handle held straight up and down in the palm, thumb curled around the handle, looks akin to making a fist.
Reverse grip/icepick grip: similar to hammer grip, but with blade pointing down; a variation on the theme has you cap the pommel with your thumb.
As for why a "saber grip" is called a "saber grip", that seems to stem from several types of cavalry sabers which were designed for giving point and had a dished out indentation on the back of the handle for your thumb. This made it easier to bring the weapon "on line" with it's target.
You seldom use this grip with a saber when fighting on foot, except to thrust, depending on handle configuration.