When learning to use awk, I found the expression awk 'c&&!--c;/regex/{c=N}' as a way to select and print the Nth line below the line matching the regex. I understand that c is initially equal to 0 so the first match isn't printed but despite having searched high and low, I don't know how to interpret the remaining syntax (outside of the /regex/) and how it specifically knows to count N lines before printing.
Can someone explain what c&&!--c means and how it works as a counter with the rest of the function?
The first half of the expression c will evaluate to true if c != 0 as you correctly guessed.
The second half of the expression !--c will evaluate to true if --c evaluates to 0; this happens when c==1 immediately beforehand. Moreover, the expression will always decrement c as long as c != 0, so c can serve as a line counter.
When the regular expression matches, we set c == N so that after exactly N lines (each one decrements c by 1), c==1, and awk will print the line.