After Eros was discovered and named in 1898, a convention was started whereby asteroids with odd orbits were given masculine names. Eros, the asteroid, is approximately twice the size of Manhattan. NEAR, which on Earth weighs 1,775 pounds, is the first spacecraft to ever orbit a small body - it has temporarily become a satellite of Eros. Complicating matters, Eros spins quite rapidly - rotating once every 5 hours and 17 minutes. So NEAR will need to be tweaked to keep it from crashing into the surface of Eros, which for the less romantically inclined is also known as asteroid 433.
Gravity depends upon both mass and distance - of course, it decreases with distance.
It does not affect an object's mass, the amount of matter a body is composed of, but how much one weighs depends on the mass of a body and the strength of its gravitational field. Thus, weight is a measurement of the pull of gravity on an object and will change with a change in the objects location in space. Mass is constant regardless of the location. If a planet has a moon, by observing the time it takes for the moon to orbit it and utilizing Newton's Law of Gravitation, the planet's mass can be determined. Similarly, a spacecraft in orbit around an asteroid can assess its mass by measuring its gravitational field.