The calculation for escape velocity is a pretty simple conservation of energy problem.
K at infinity =.5mv2 = 0 because v at infinity = 0
U at infinity = GMm/r = 0 at infinity because r = infinity
.5mv2 = GMm/r
From there it's simple algebra, and escape velocity is ve = sqrt(2GM/r)
This equation's applications are seen in the exploration of space. Spacecraft need to reach escape velocity in order to not eventually crash back into the earth's surface. Some satellites are orbiting earth at just above escape velocity, meaning that they are actually spiraling away from the planet. On the other hand, some satellites are orbiting just below escape velocity, meaning that they will eventually fall into the atmosphere and burn up. However, some of these satellites have on-board rockets which can change their trajectory, allowing for more stable orbits and longer lifetimes.
The Voyager 1 spacecraft used its escape velocity to leave the solar system and explore what lies beyond.
NASA's Curiosity mission required the spacecraft to reach near escape velocity (although I'm sure the actual spacecraft reached a higher speed) to make it to Mars.
As humans explore more of the space that surrounds the planet, escape velocity and its applications will become even more important.