"The good rising fastball is the best pitch in baseball." - Tom Seaver
A fastball is the simplest of all pitches, a ball thrown for speed. Athough the fastball is not considered to be a breaking pitch, the most effective fastballs will have some movement. Fastballs are gripped with the index and middle finger and the thumb, but the exact movement depends on the details of the grip.
A four-seam fastball is gripped with the fingers across the seams of the ball. Because the fastball is thrown with backspin, it gets some lift and won't drop as much as a smooth ball (or one with little spin) would. A four-seam fastball doesn't get enough lift to rise, but it may look as though it rises or "hops" to the batter. The movement is just enough to make the batter swing under the pitch for a fly ball or pop-up, or even to miss the ball completely.
A two-seam fastball is gripped with the fingers along the seams. It doesn't get as much lift as the four-seam fastball, which makes it drop more than the batter expects. That makes the batter tend to hit the ball downward for a ground ball. The difference in grip makes a two-seam fastball slightly slower than a four seamer.
A cut fastball (aka cutter) is gripped asymmetrically, either by holding the ball off-center, or by gripping harder with one finger than the other. The asymmetric grip gives the ball some sideways spin in addition to its backspin, resulting in sideways movement similar to, but less than, a slider. The small sideways movement isn't enough to make the batter miss the ball, but may make him hit it away from the "sweet spot" of the bat for a weak hit and sometimes a broken bat.
A split-fingered fastball (aka splitter) isn't really a fastball at all. It's thrown with a fastball arm motion, but with the fingers spread wide apart to neutralize the ball's spin.