A curveball (aka curve, hook, yakker, Uncle Charlie and The Hammer) is a breaking ball with less speed and more movement than a slider. A curve is thrown with the forearm rotated 90° inward, and often, though not always, with a deliberate snap of the wrist as the ball is released.
For decades, it was debated among fans whether a curveball really did curve. Fans would sometimes set up a series of posts, and get curveball pitchers to throw balls past the posts to see the amount of curve, if any. Applied physicists finally settled the question that yes, it does curve.
A curveball is always slower than a fastball, but the degree of difference varies widely. Some pitchers throw a hard curve that is almost as fast as a slider, while others throw a slow curve that serves effectively as a change of pace. The curve can be thrown from any arm angle, with the ball's breaking in the direction of the pitcher's arm movement.
An overhand curve, drop curve, or 12 to 6 curve is a ball thrown with an arm motion directly over the pitcher's shoulder. Because its motion is straight down, the overhand curve is equally effective against left and right handed batters.
A 3/4 curve, roundhouse curve, or 11 to 5 curve is thrown from a 3/4 overhand position. It drops and moves slightly away from a batter with the same handedness as the pitcher or toward an opposite handed batter.
A sidearm curve or outcurve is thrown from a straight sidearm angle and breaks directly away from a same handed batter or toward an opposite handed batter. A well thrown sidearm curve is terrifying to batters with the same handedness as the pitcher; it can look as though it's going to hit the batter and then break enough that it crosses the outside of the plate. The sidearm curve is a classic LOOGY pitch, since it's much more effective against same handed hitting.
An underhand curve is thrown underhanded as the name implies. It doesn't break straight up because the spin on the ball isn't enough to overcome gravity, but it does drop less than expected.