Counting down the seconds leading up to a rocket launch is iconic - certainly it seems that every child knows that after "5, 4, 3, 2, 1" comes "Blast off!" However, the countdown can't always have been associated with rocket launches. My impression is that rockets originated as weapons, and I have a hard time imaging the countdown in the midst of a heavy battle. Therefore, when did rocket launches first start using a countdown, and why did it become so well-known?
According to Rockets, Missiles, and Space Travel by Willy Ley, the Fritz Lang movie Girl in the Moon (1926) featured a fictional rocket launch countdown for dramatic purposes and it was adopted by the experimenters in the German Rocket society for thier launches. Werner Von Braun and others became engineers in the German military rocket program and then in the US military and civilian rocket programs and kept on using countdowns.
NASA has always had someone announce milestones in the countdown, here's Mercury 3 (first manned flight with Alan Shepard, around 0:30) for example. And Explorer 1 (around 2:26), the first US satellite.
I'm not sure about the Russian space program, but the Vostok 1 launch doesn't seem to have an audible 10-0 countdown.
Before that, most of the V-2 launch footage I've seen was silent or overdubbed with later commentary, so no idea if the Germans used a countdown.