The drop kick was often used as a surprise tactic. The ball would be snapped or lateraled to a back, who would perhaps fake a run or pass, but then would kick the field goal instead.
This method of scoring worked well in the 1920s and 1930s, when the football was rounder at the ends. Early football stars such as Jim Thorpe and Paddy Driscoll were skilled drop-kickers.
In the 1930s, the ball was made pointier at the ends. This made passing the ball easier, but made the drop-kick obsolete, as the pointier ball did not bounce up from the ground reliably.
In Australian Rules football, a similarly named and executed kick was used in general play, particularly after a free kick was awarded. It was popular as players could kick the ball long distances, and the ball's backwards rotation was reasonably easy for teammates to catch (a major feature of the game).
A variation known as the "stab pass" or more poetically, the "daisy cutter" involved an abbreviated follow-through and travelled on a notably low trajectory, which made it very useful for short-range passing.
The drop kick and stab pass gradually disappeared from the game by the 1980s, as it was unreliable, particularly on wet grounds, and players were coached to always use the drop punt kicking style to avoid having to make a decision on what kind of kick to perform.