I just have a quick question about time dilation/proper time because my physics book makes it a little confusing. Let's say we have an observer on Earth, and then an observer on a space ship. The space ship leaves Earth, flies to the Moon, and then returns to Earth. Who is the person measuring the proper time and why? I know that a clock "runs slower" when it is in motion because it is in frame S' which is the rest frame of the clock, but doesn't the observer on Earth also have a clock that is in it's rest frame?
Each observer has their own proper time measured by the clock in their rest frame. However, one man's proper time is not another man's proper time. Time dilation means that each observer will see the other observer's clock running slower (compared to their own proper time measuring clock). But everything is perfectly symmetric from either observer's point of view as long as the relative motion is uniform. You measure your clock ticking at the "normal rate" (your proper time) and you see the other person's clock ticking at a slower rate. Similarly the other person measures their clock ticking at the "normal rate" (their proper time) and they see your clock ticking at a slower rate.
This is all well and nice, but it gets interesting when the two compare their clocks after one of them does a round trip. This means that one of them necessarily had to accelerate and decelerate and was not in uniform motion (technically, was not on a geodesic). Now you have an opportunity to actually compare those two clocks and you'll always find that the person in uniform motion (in this case, the observer at rest on Earth) was the one whose clock has ticked the most, and hence aged the most.
The best way to understand this is to realize that the length of paths in spacetime is measured by the total proper time along that path (measured by that path-traveller's clock in their rest frame). One can show that the paths of uniform motion (geodesics) have that length maximized, so any path that deviates from a geodesic (because of accelerations), will necessarily measure a shorter total proper time after a round trip.
EDIT AFTER FIRST COMMENT: Time dilation isn't the appropriate effect to consider in this particular problem -- length contraction is. In Nick's frame, a length contracted ship passes by at speed $v$. In Molly's frame, a point-object (heh) Nick passes by an uncontracted ship at speed $v$. Clearly, this should happen quicker in Nick's frame because of the length contraction. Thinking in terms of time dilation simply doesn't help here. Think from the point of view of each observer and it will be quickly obvious which effect to use.