escape velocity
Throw a ball up and it comes back. Throw it harder and it goes higher before returning. Is there a speed so great that it never comes back at all? Yes, and it is called the escape velocity: the minimum launch speed that lets an object coast away from a planet forever, gradually slowing but never quite stopping, without any further push from an engine.
It comes straight from energy conservation. To just barely escape, an object's kinetic energy at launch must equal the depth of the gravitational potential well: 1/2 m v^2 = G M m / r. The mass m cancels, giving v_esc = sqrt(2 G M / r), where M is the planet's mass and r is your starting distance from its centre. Notice it is exactly sqrt(2) times the circular orbital speed at the same radius. For Earth's surface the value is about 11.2 km/s, roughly 40,000 km/h.
Two honest points. First, escape velocity does not depend on the mass of the escaping object; a pebble and a spaceship need the same speed. Second, it is a speed, not a velocity in a fixed direction: as long as you are not aimed into the ground, any direction works (ignoring air resistance and the pull of other bodies). Real rockets do not actually reach 11.2 km/s in one instant; they burn fuel continuously, so they can climb away more gently instead of being flung all at once.
Earth's escape velocity is v_esc = sqrt(2 G M / R) = sqrt(2 x 6.67 x 10^-11 x 5.97 x 10^24 / 6.37 x 10^6) = about 11.2 km/s. The Moon's, with far less mass, is only about 2.4 km/s, which is why it was easier for the Apollo crews to lift off from there.
Less massive worlds have smaller escape speeds; the escaping object's own mass never matters.
Escape speed is independent of the escaping object's mass and of launch direction (ignoring air drag); with air resistance you need somewhat more.