How can one extra digit catch a typo or a flipped bit?

By adding a little redundancy: extra information the receiver can check against the rest. The simplest version is the parity bit. Before sending a group of bits, you add one more bit chosen so that the total number of 1s is even. If a single bit flips on the way, the count becomes odd and the receiver knows something went wrong. Your credit card number plays a similar trick: its last digit is a check digit computed with the Luhn algorithm, which catches every single mistyped digit and almost every swap of two neighbouring digits.

Detecting isn't fixing, though. Parity can't tell you which bit is wrong, so the data has to be sent again, and if two bits flip, the count looks fine and the error slips through.

That limitation drove Richard Hamming mad. In the late 1940s at Bell Labs, his relay computer would stop and flash its lights for an operator when it found an error, but on weekends, with nobody there, it simply abandoned his job and moved on. "If the machine can detect an error," he fumed, "why can't it locate the position of the error and correct it?" His answer, the Hamming code published in 1950, uses several overlapping parity bits so that the pattern of failed checks points straight at the broken bit. It still protects computer memory today.