A balk occurs when a pitcher makes an illegal motion on the mound that the umpire deems to be deceitful to the runner(s). As a result, any men on base are awarded the next base, and the pitch (if it was thrown in the first place) is waved off for a dead ball.
Exactly what constitutes a balk is summed up in section 8 of the MLB rules, which describes a legal pitching delivery.
Pitchers with high balk totals are also generally adept at picking runners off, this being because their moves to first base are typically so deceptive that they border on being illegal. Any umpire, if he notices an illegal movement by the pitcher, can call a balk.
The specific rules for balks were first introduced in 1898 to prevent pitchers from intentionally deceiving baserunners. Without balk rules, pitchers had any means of fooling baserunners, who had to act conservatively on the bases as a result.
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