The kv value is a rating of how many rpm the motor makes per volt, at no-load. A 1000 kv motor, with 10 volts applied to it, would spin at 10,000 rpm if it didn't have a prop on it. When you add a prop to the motor, the prop loads down the motor, and actual rpm is lower, but all else being equal (prop size and voltage), a higher kv motor will spin faster than a lower kv motor.
The current that a motor will draw is proportional to its speed (rpm's) and the size of the prop that's on it. A bigger prop will move more air and make the motor draw more current. A higher rpm will do the same. Therefore, you can see that there is a sliding scale wherein a low-kv motor with a big prop, and a high-kv motor with a small prop may draw the same current.
Here's the cool part: a prop's thrust is proportional to its area, not its length. And area scales as the square of length (again, this is a simplification, since the shape of the prop is not fixed, but go with it). What this means is that a long prop being pushed by a low kv motor will be more efficient (generate more thrust-per-watt) than a small prop being pushed by a high kv motor.
This is why, for heavier lifting, you tend to see low-kv motors with larger props, whereas for lighter copters, you tend to see high-kv motors with smaller props.