In rocketry, a brachistochrone trajectory is the path between two orbiting bodies (such as Earth and Mars, for example) that requires the shortest transit time. Unfortunately, a brachistochrone trajectory also requires by far the most delta-v.
To perform a brachistochrone transfer, a spacecraft would thrust continuously until the midpoint of its journey, then flip around and thrust in the opposite direction until it reaches zero velocity relative to its destination. At this point a final thruster burn would be required to enter orbit. This trajectory is so named because the path a spacecraft follows as it flies like this is a mathematical brachistochrone, as described in bitter_engineer's rabbit and dog example above.
Though brachistochrone transfers are very fast, they also require impractically large amounts of delta-v. A brachistochrone transfer from Earth to Mars at 0.01 g, or about 0.1m/s^2, requires 375 km/s of delta-v. By way of comparison, a Hohmann transfer from Earth all the way to Pluto requires only 25 km/s. This is in turn more than was available to the longest-ranged manned mission in history, the Apollo missions to the moon, which carried about 14 km/s of delta-v (and required a monstrous mass ratio of 22 to do it).