Baseball, by the Rules

Pitcher Valente Bellozo (of the Miami Marlins) balks by dropping the ball while standing on the mound, before a plate appearance against Brent Rooker of the Sacramento Athletics.
Author's Note: MLB's video offerings, while very fine in many respects, do not include balks, or really any hint of a balk. Go ahead, try to find a balk! You can't! Where are the balks?
Balks
Balks are… complicated, but here goes. The simplest version would probably be to say that balks are a catch-all term for a set of illegal motions, made by a pitcher, usually with the intent of trying to fool a baserunner that they would very much like to pick off.
So, what constitutes an “illegal motion” in this sense? Everything starts with the pitcher. Before the pitcher goes into their motion, they must first “come set,” which means being posed perfectly still, on the pitching rubber, with their hands together. After coming set, the pitcher can go into their normal pitching motion, or attempt a pick off. If they don’t fully come set, or if they flinch1 before going back into motion, it’s a balk. Once they go into motion, they can’t go back to the set position unless they step off the pitching rubber, and once their hands come apart, they must deliver either a pitch, or a pickoff. Violate any of these rules, and it’s a balk2.
Finally, when the pitcher does go to throw the ball— either as a pitch, or as a pickoff— they must step towards whatever target they are throwing to. So, no striding towards home plate, then delivering a pickoff throw to first. This is easily the trickiest part of the balk rule; there’s a lot of nuance about what really counts as stepping towards a base, especially if you’re a left-handed pitcher who is facing first base when you come set. In reality, umpires could probably call a lot more balks than they actually do3.
There are a few other ways to do a balk— like straight up dropping the ball, for example. There’s a lot of nuance to all of this, and if you’re just starting out with baseball, it’s probably not worth getting too in the weeds. (If you really do want more details, this article from MLB.com can help you out.) Even the most veteran baseball observers have trouble with balks!
When a balk is called, any runner currently on-base gets to advance by one base, regardless of whether or not they are forced to. This includes a runner on third advancing to home and scoring. When an umpire calls a balk, they will generally do so by holding their hand up in a sort of “wait a minute” gesture, before motioning any affected runners to their next base. A balk call negates anything else that might have happened— if the pitcher pitched, it’s nullified, if they attempted to pick the runner off and succeeded, it doesn’t count. Play resumes, and because the runner(s) has/have moved up, the pitcher gets a new set of disengagements to work with.