According to Newton's Universal Law of Gravitation, in order for an object to remain in orbit there must be a "cetrifugal" force balancing the gravitational force. He expressed this relatioship in his orbital equation
Fg = Fc where Fg is the gravitation force and Fc is the centrifugal force.
GM1M2/r^2 = M2V^2/r
Solving for V gives
V = sqrt (GM1/r) So, in order to maintain an orbit at a distance of r above the center of the Earth, the object must maintain an orbital speed of V given here.
This is different from escape velocity. Escape velocity is the minimum vertical speed required to over come the gravitational pull of the Earth and to leave its orbit. Escape velocity for the Earth is relatively constant and is approximately 7 miles/s.
Other planets in theory effect the escape velocity in that every object with mass provides a gravitational force on every other object with mass. So Jupiter is pulling a rocket trying to leave the Earth and is therefore reducing the force the rocket itself has to apply to achieve escape velocity. However, the distance between the Earth and other planetary bodies is so large that the effect is negligable.