Earth's gravitational force of attraction vanishes at what distance?
This might seem intuitively wrong because on Earth, we simply have little or no experience with things that exist over an infinite distance. It also requires an introductory understanding of particle physics and forces.
There are four known forces which are considered fundamental, meaning that they cannot be simplified any further, and they explain all known force effects. Gravity is one of these forces. Forces exert their influence on other objects through what are called bosons, or "force-exchangers". The range at which a boson can exert its influence is determined by its mass. Therefore, big heavy bosons, like those which operate inside the nucleus of an atom, are very short-range forces because the bosons are very heavy. This explains why humans never personally experience the enormous forces involved in holding a nucleus together, but we can utilize these forces in the form of controlled nuclear reactions.
Gravity and Electromagnetism are the two fundamental forces that are visible to us in our everyday lives. This is because their bosons are massless, meaning that they can travel for an infinite distance, but their corresponding effect is relatively weak.
Part of the reason for this weakness is that both gravity and EM follow the inverse square law. This means that the force being felt gets exponentially smaller the farther you are from the source. Therefore, a human on the moon would already be feeling very little of the Earth's gravity, and by the time you got to the edge of the solar system you'd feel almost none. But no matter where you go in the universe, you would always feel some of the Earth's pull.
There IS a way to get outside of the range of a gravitational effect. Gravity appears to travel at the speed of light. If you were to somehow teleport yourself to a location that is about 5 billion light years away, the gravity of the Earth would not yet have reached that location, and you would therefore have found a point at which the force of attraction "vanishes".