Relativity Confusion In the time dilation formula $\Delta t=\gamma\Delta t_0$, I am confused about what $\Delta t$ and $\Delta t_0$ measure. 
If we have two people, Tom and Bob, moving at constant velocity relative to each other, and we say Bob is at rest while Tom is moving close to the speed of light, Bob will measure time in his own frame and will think that Tom has recorded less time. 
We can switch the roles and have Tom at rest and Bob moving close to the speed of light. Tom will measure time in his own frame and will think Bob has recorded less time. 
Do both $\Delta t$ and $\Delta t_0$ measure the change in time measured by each guy in their own frames or do they measure the change in time that each guy thinks the other guy has measured?
 A: $\Delta t_0$ is the time between two events that occur in the same place in an inertial frame of reference, as measured in that frame. The events might indeed be two ticks of the same clock, 'C'.
In another inertial frame of reference, the same events will occur in different places (for example, in this frame, C will be moving and the ticks will happen in different places). In this new frame, two (synchronised) clocks will be needed to time the interval between the events, because the events are in different places! The interval as measured using the two clocks in this frame is $\Delta t.$ 
The postulates of Special Relativity (or, if you prefer, the inter-relatedness of time and space) imply that $$\Delta t=\gamma \Delta t_0.$$ The key thing to note is that $\Delta t_0$ is a special time (a 'proper' time) because it is the time between two events as measured in the reference frame in which the events occur at the same point in space.
I don't want to be rude about Tom and Bob, but I don't think that what you've said about them captures the essence of time dilation. [And the slogan "Moving clocks run slow" is of little use unless you know how to interpret it!]
A: You are not wrong to have this confusion.
When two people are moving relative to each other, and they both make their best guess about what is happening inside of the other person's spaceship, they both think that the other person is moving in slow motion.
Early in relativity this was thought to be obviously self-contradictory. Today with modern technology we can maybe more clearly understand why: “OK Dr. Fancy Physicist, I hear what you are saying, but surely if Alice thinks that Bob is moving in slow motion and Bob thinks Alice is in slow motion, surely we can resolve this: just have Alice and Bob start talking on a cell phone! Surely one of them will actually be getting their words out faster than the other one and we will know who is right and relativity will be disproven!”
Well, not so fast. How does a cell phone work? It transmits microwave pulses. How fast do they go? The speed of light. It turns out this finite propagation speed eats up any possibility for Alice or Bob to figure out who is right. 
In relativity it is best to define events: these are points in spacetime, so they are a combination of a place in space and some instant in time. Like maybe one event is a star going supernova, a big explosion where the star suddenly becomes very bright.
Coming out of any event, there is an expanding light bubble that tells the entire rest of the universe that this event has happened. Everyone sees this light bubble as perfectly round and moving at the speed $c$. Fast forward until some other event happens: it might either be located inside the bubble, in which case it already knows that the first event has happened, or on the bubble, or outside the bubble. Then it has its own bubble which expands. In the first case, the second event's bubble is contained within the first event's bubble, in the second case the two share one point, but otherwise the second event is clearly “inside” the first one, and in the third case the two bubbles start off not overlapping and eventually they overlap on a circle after which they will always overlap on a circle.
We say that in the first case the two events are objectively ‘time-separated.’ Everybody will always see the one bubble inside the other bubble. In the second case, we say that they are ‘null-separated.’ And in the third case, we say that they are ‘space-separated.’
Two things that are objectively time-separated are not objectively separated by space. I can prove it: a spaceship cannot go faster than light, but it can hypothetically go any speed less than that. Since the one light bubble begins in the other one, you can imagine a spaceship that flies from the one event to the other event, without accelerating: for such a spaceship, both events happened “right here, outside my window.” For them, both bubbles are perfectly centered, one is perfectly inside the other. And here's the important thing: this spaceship measures an objective time delay between the two events, the smallest time delay that anyone sees, which is called the proper time between the two events. Everyone else sees a longer time.
For the thought experiment with Alice and Bob, notice that they are both talking about watch-ticks, these are events that are at rest in Alice's or Bob's spaceships but are moving in the other person's space ship. The one who sees the watch stand still in space, measures those ticks in proper time. Everyone else sees the watch ticking slower.
The converse is also true: it turns out that whenever you accelerate in relativity, all of the clocks ahead of you tick faster and all of the clocks behind you tick slower, in proportion to both your acceleration and their distance from you. This continues until you stop and they all come to some fixed offset. It turns out that this is the only new first-order prediction of relativity, you can derive length contraction and time dilation from this plus old Newtonian physics plus some matrix mathematics. But because of this, you can resize these expanding light bubbles by accelerating one way or another. If two events are spacelike separated there is some spaceship which thinks that both light bubbles are the same size, so that both events were simultaneous, and then by accelerating in one or another direction before both events, some spaceships may think that instead one or the other event happened first. They are objectively space-separated but not objectively time-separated. And the spaceship which sees both bubbles at the same size measures an objective length between the two events which is the longest possible such length; everyone else measures a shorter length due to length contraction. 
I’ll not get into the null-separated events except to say that they are a bit weird because they are objectively both space and time separated but either their space or time separation can be brought arbitrarily close to zero, but not quite all the way there. If you are an experimentalist then you can almost always pretend for real events that are not causally connected by some transmission of light that they are either one or the other, within measurement error. If you are a theoretician then things might get much more strange if you have a wild imagination, for example one might imagine that actually every particle moves at the speed of light all the time but that we are seeing some sort of quantum “slowdown” for massive particles that is not dissimilar from how light slows down in glass or water. 
