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 three-quarter 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.
"Well, this year I'm told the team did well because one pitcher had a fine curve ball. I understand that a curve ball is thrown with a deliberate attempt to deceive. Surely this is not an ability we should want to foster at Harvard" - traditionally attributed to Harvard University president Charles Eliot
As the above quote, popularized by Ken Burns in his documentary Baseball, shows, the curve ball was controversial when it first appeared on the scene. The quote is genuine in that it does accurately reflect the views of some persons at the time, but its attribution is incorrect. It was in fact Humanities professor Charles Eliot Norton, a cousin of Charles Eliot, who pronounced it, and he was an outsider to the game, his views being one of a reactionary remembering the days of his youth four decades earlier when pitchers were still lobbing balls towards the batter to let him hit it as hard as possible.
- David Adler et al.: "These are the nastiest curveballs in baseball", mlb.com, March 30, 2020. 
- Dave Baldwin, Terry Bahill and Alan Nathan: "Nickel and Dime Pitches", in The Baseball Research Journal, Society for American Baseball Research, Cleveland, OH, # 35 (2007), pp. 25-29.
- Richard Hershberger: "'With a Deliberate Attempt to Deceive': Correcting a Quotation Misattributed to Charles Eliot, President of Harvard", Baseball Research Journal, SABR, Vol. 46, Nr. 1 (Spring 2017), pp. 65-69.