The continue countdown screen was introduced to arcades in the 80s. It was a simple and effective way of getting players to fork over more money. When you fail, the screen comes up for a limited time with a countdown to insert additional coins or press a button.
Most of these games' continue screens were pretty basic—they used simple text or huge flashing numbers on a black background—but there were some really cool exceptions, like the one above from Cadillacs & Dinosaurs and like these below:
Street Fighter II
Mortal Kombat 4
Final Fight 3
A lot of non-arcade games experimented with unique continue screen designs, too, nixing the traditional (and now useless) timer:
Metal Gear Solid
There are many more! You should storm the comments with your favorite picks.