How are gravity tractors possible? I've heard about how NASA is planning to redirect asteroids by having a satellite tug on it gravitationally using a large enough bolder, but how is that possible? 
I thought that changing the trajectory of something as big as a mountain would require a lot of kinetic energy and I don't see how a tiny satellite can have that much kinetic energy.
 A: 
I thought that changing the trajectory of something as big as a mountain would require a lot of kinetic energy and I don't see how a tiny satellite can have that much kinetic energy.

A gravity tractor would require lots and lots of time, and most likely involve lots and lots of little satellites.
Suppose we knew that an asteroid would hit the Earth if we did nothing about it, and suppose that we knew that several decades in advance of the predicted collision date. With that kind of lead time, changing the asteroid's velocity by a centimeter per second or so would suffice to make the asteroid miss colliding with the Earth. An acceleration of 10-10 m/s2 applied continuously over the course of a decade would more than do the trick.
This tiny acceleration is well suited to vehicles with high specific impulse thrusters. A tiny amount of thrust would suffice to keep the vehicle from falling into the asteroid. Keeping the vehicle relatively close to the asteroid would pull the asteroid behind the vehicle thanks to gravitation. Should vehicle #1 run out of propellant mass prior to achieving the desired delta-V, replace it with vehicle #2. Rinse and repeat.
What if we don't have several decades of advance notice? The answer is simple: This approach won't work. We'll need to use something more drastic.
A: An alternative technique to Electro's answer, although with the same name, as described on Gravity Tractor.
This is done in 4 stages: 


*

*Guide a spaceprobe, which is fitted with a claw arrangement, to the asteroid that requires it's orbit redirected and land on it.

*Pick up the largest boulder that fits into the claw, as illustrated below.
 


*Now lift off the asteroid, having  increased the mass of the spaceprobe by the mass of the attached rock, and "hover" over a point on the asteroid, deflecting it by continually pulling on it.

*Given enough time, the tiny gravitational pull of the spaceprobe will affect the orbt of the asteroid. This is on a timescale of years.
