Gravity is a force between objects/particles proportional to the objects' mass. Newton's universal gravitation looks like this:
Fg = - G m1m2/r2
where G is a proportionality constant, the m's are the masses of the two objects in question, and r is the distance between the two objects. This is why we feel the Earth's gravity affect us, but we don't feel the moon's or sun's gravity affect us. They most definitely influence the Earth (since the sun causes our orbit and the moon causes the tides), but we don't feel the effects of their presence.
So, if we have an object with mass m on Earth in free fall, its equation of motion is determined by
Fg = m a = - G m ME/r2
where ME is the mass of the Earth and a is the acceleration of the object. Note that, if we divide both sides by m, we find that
a = - G ME/r2
which means that the acceleration of an object in free fall has nothing to do with the mass of the object. In fact, you can see a video of this on the moon at Wikipedia's Gravitation page that shows Apollo 15 astronaut David Scott dropping a feather and hammer simultaneously. Since there is no air on the moon, the feather is not afloat longer than the hammer, and they fall at the same rate and hit the ground at the same time.
Also, while I said earlier that gravity affects things with mass, it also affects light, which does not have (rest) mass. However, light has energy, and as Einstein showed with his Special Theory of Relativity, energy and mass are equivalent:
E = m c2
So, you can construct the relativistic mass of light, thereby finding the equations that govern the changing of the straight path of light in a gravitational field. Using Einstein's General Theory of Relativity, you can also view the gravitational field as a curvature of spacetime, which influences straight lines to be curved in the space near the massive object, affecting the path of light.
Another interesting thing about mass: objects actually have two different masses associated with them: gravitational mass and inertial mass. Gravitational mass tells you how much an object interacts gravitationally, while inertial mass tells you how much an object resists a change in motion. In other words, more massive objects take more force/energy to alter their paths than objects with less mass. Here's the interesting thing, though: both these masses are equal, even though there really is no physical law stating that they have to be. The only reason we know these masses are equal is because empirical evidence says they are; there is no indication that these two masses are different to an appreciable/statistical extent.
So, if you think that there are no unanswered questions in the realm of physics, you are sorely mistaken.