Halley's method requires one to measure the timing of the beginning of the transit and the end of the transit; both pieces of data have to be measured at two places of the Earth's globe whose locations must be known.
The picture by Vermeer, Duckysmokton, Ilia shows that the two places on Earth have differing locations in two different directions (the differences in the distance from the Sun and Venus are too small to be measurable): one of them is parallel to the direction of the transit of Venus and will be reflected in the overall shift of the timing; the other component is transverse to it and it will actually shift the line along which Venus moves and crosses the Sun in the up/down direction i.e. it will make the duration of the transit longer.
Each of these pieces of data – overall shift in the timing arising from one coordinate's difference between the two terrestrial locations – and the difference between the length of the transit – due to the other coordinate – are in principle enough to determine the solar parallax. Because synchronization of clocks at very different locations was difficult centuries ago, I suppose that the latter – the difference between $\Delta t_1$ and $\Delta t_2$ – was probably more useful historically. But we're talking about O(10) minutes differences in both quantities.
The calculation of the parallax from $\Delta t_1$ and $\Delta t_2$ and their difference is a simple exercise in geometry but I want to avoid trigonometric functions here.
At any rate, Halley didn't live to see a proper measurement (the transit occurs about twice a century and the two events are clumped together with a 8-year break in between). The best he could get was 45 angular seconds for the parallax; the right answer is about 8.8 seconds. He knew that his result was very inaccurate. Note that the solar parallax is the angle at which the Earth's radius is seen from the Sun, i.e. the difference in the rays needed to observe the Sun from the Earth's center and/or a point on the Earth disk's surface.
When you convert 8.8 angular seconds to radians, i.e. multiply by $1/3,600\times \pi/180$, you get $4.3\times 10^{-5}$. Now, divide 6378 km by this small number to get about 150 million km for the AU.
Some orders-of-magnitude estimates for the numbers. Venus orbits at 0.7 AU so it's actually closer to the Earth than it is to the Sun during the transit. It means that a shift by 6,000 km up/down on the Earth's side corresponds to about 12,000 km up/down on the Sun's side. So the two horizontal lines crossing the Sun on the picture (places on the solar surface where Venus gets "projected") may be separated by about 12,000 km. Compare it with the solar radius near 700,000 km: you may see that we're shifting the horizontal lines by about 1% of the Sun's radius and the relative difference between $\Delta t_1$ and $\Delta t_2$ will be comparable to 1%, too. The last transit in 2004 took about 6 hours so the difference in the duration at various places is of order 10 minutes.
The 2012 transit of Venus on Tuesday night UTC will take over 6 hours, too; the timing and duration differs by about 7 minutes depending on the location, too.
If you've been dreaming about observing the transit of Venus, don't forget about Tuesday 22:49 night UTC; the following transits will occur in 2117 and 2125. There is a blog version of this answer, too.
The excitement behind various claims is somewhat excessive.
First, the Mayan astronomers, see e.g. Mayan astronomy at this page, didn't use any armillary spheres or sextants as others did. Their observations were made with naked eye and they were depicting positions of planets with crosses. The accuracy of the Venus' position after a synodic 584-day cycle was off by 2 hours.
It's good but it isn't unbelievably accurate in any sense. It just means that the moment when the Venus "returns to the original place" was measured with a 2-hour accuracy. As far as I know, all the Mayan models for the orbits were periodic and the imagined trajectories were circular. They have just played with lots of periods which is why they ended with lots of these baktuns and pictuns (cycles) and parallel calendars. But it was numerology based on the known periodicities, not really a precise framework to predict the positions. Let me also emphasize that there was no "dogmatism" concerning the identity of the "center of the Universe".
The Babylonians did all the astronomy with the tables. Lots of tables. Of course, they could do things much more accurately than the Mayans. It was a phenomenological approach with a lot of hard work whose basic logic only depended on arithmetics applied to measured data adjusted so that the agreement with all the data is reached. They were mapping the trajectories of the Sun and the planets in some coordinates, observed some periodicities, and were ready to calculate all the required corrections by some arithmetic formulae that were simply a good fit.
They didn't have any prejudices about the circular shape of any orbits, centers of the Universe, relationships between planets and cosmology (birth of the Universe), and so on. It was a heavily empirical approach. You could say that it was modern science but you would be missing the fact that they were simply not looking for the laws of physics that explain all the patterns and they were not attempting to reduce the amount of the independent mess. The mess was immense, the trajectories looked rather general. Effectively, the success and precision of their predictions for the locations boiled down to their having very complicated "laws of physics", laws with tons of (measured) corrections and special rules for individual planets and individual situations.
That's why the Greek astronomy represented dramatic progress because they were actually trying to make things simple, to see a rational explanation of the causes behind the particular trajectories. They were not satisfied with the empirical side of the story and that's why it was such a breakthrough. They were building mechanical models. When it comes to the accuracy, whether the orbits were thought of in the geocentric or heliocentric frame is an irrelevant technicality. The heliocentric frame is closer to allowing us to write Newton's equations as the full classical explanation which is a "culmination" of non-relativistic celestial mechanics. However, one has to do lots of other things before these final steps in order to represent the observed positions on the two-sphere by some actual trajectories in the 3D space which have a cause.
So the Babylonian models were "overfitted", using an immense amount of observed data that weren't independent from each other. Greeks wanted simple enough models that account for the facts. Circular motion is the most symmetric one so they started with it and tried to match the observations in various ways. It didn't work quite exactly so they began to build the epicycles (smaller rotating circles) on top of the deferents (main circles that carry the objects).
Epicycles are often talked about in a negative light but they represented (and similar methods represent) a very progressive intermediate phenomenological description in between the "Babylonian tables" and "simple models of trajectories given by simple equations". The idea that one circle carries another which carries the planet is pretty much equivalent to some kind of a Fourier expansion of $\vec r(t)$ and the Greek astronomers were picking several leading coefficients of this expansion (by observing them). I am sure that most people who talk dismissively about epicycles wouldn't be capable of doing anything of the sort – and anything superior, either.
Ptolemy's model was finally a very accurate description. It was a geocentric model – not only the Sun but also the other planets (and, correctly, the Moon) – were directly orbiting around the Earth. But that didn't hurt the accuracy because the orbits were described in such a way that the Sun-Earth separation was correctly subtracted, anyway.
Now, note that if the exact distances of the planets were known one could notice that in the case of all planets, we are suspiciously subtracting the same periodic function with the period of 1 year, namely the Sun-Earth separation, and we could figure out that in the heliocentric frame, things get more natural. However, when you don't know the absolute distances, things are a bit tougher. They were not aware of the possible task "get simpler by going heliocentric" so they didn't pursue it. It wasn't a problem. Even in Newton's physics, we may describe things in the Earth's frame (even a spinning, non-inertial frame) as long as all the fictitious forces are correctly added.
It's another myth when people believe that the heliocentric model implies that it has no epicycles. Instead, the Copernical model had an even greater number of epicycles than Ptolemy's model because some of the new ones replaced the equants. Ptolemy's equants were actually a rather clever observation that played a similar (although not quite as accurate) role as Kepler's observation that the Sun sits at a focus of the elliptical orbits. Of course, Kepler's precise observations about the ellipses were not known to Copernicus and they don't directly follow from heliocentrism.
Best Answer
There's actually at least one very big clue that's been accessible to skygazers since the earliest times: the first quarter moon at dusk. Every child in the northern hemisphere going back to 30,000 BCE likely would have been familiar with how 1st-quarter moons always tend to rise at noon, reach its highest point at sunset (with an azimuth directly south), and set at midnight.
Form a triangle out the observer, the sun and the moon: $\triangle OSM .$ The only angle the observer can measure directly is of course the angle between the sun and moon, the observer forming the vertex. The sun is in the direction of the horizon, and the 1st-quarter moon is near zenith, hence $\angle SOM \approx 90°.$
The angle with vertex at the moon, $\angle OMS$, couldn't be measured in general, but it doesn't take too much imagination to infer that the shape of the sunlit portion of a 1st-quarter moon results whenever $\angle OMS \approx 90°$. Hence, $\triangle OSM$ is an acute, nearly isosceles right triangle, whose legs are practically parallel and much, much greater in length than the base. This small base length is the earth-moon distance $|OM|$ is itself much greater than any terrestrial distances we measure on the Earth's surface. Thus, with extremely little effort we can be reasonably confident that Eratosthenes' condition of parallel sunlight rays holds to good enough approximation for the purpose of his measurements (uncertainties in the measurements of distances between cities would have been the limiting factor towards overall precision anyway).