Most speed cubers solve the cube in many more moves than that. The key is very high Turns Per Second and excellent look ahead and algorithm memorization.
Rubiks cube robot solvers tend to do it in fewest moves. WCA also has 'fewest moves' competitions, but i think participants can spend hours and hours on a fewest move solution
That's right. The preferred method is the Fridrich method, which consists of:
Cross - first 4 edges
F2L - first two layers, either learned algorithms (lots of them) or intuitive is easy enough with lots of practice.
OLL - orientate the last layer (all the same colour) - 57 algorithms to learn
PLL - permute last layer - 21 algorithms to learn
Lots and lots of moves. Probably at least 100 moves per solve
Most sub 20 second cubers know all the OLL and PLL algorithms and do intuitive F2L, and probably always solve the cross on the same side. This makes it easier to memorize edge corner pairings.
The BEST cubers (like Felix Zemdegs) are colour neutral (do the cross on whichever colour is quickest), do intuitive F2L with look ahead to the next edge-corner pairs (so as they pair up an edge and corner, they're looking where the next pair is), and they even look ahead to land on preferred (fast) OLL algorithms or for PLL skips (OLL finishes the solve).