What is the simplest way to prove the Earth is round? Assume you've come in contact with a tribe of people cut off from the rest of the world, or you've gone back in time several thousand years, or (more likely) you've got a numbskull cousin.
How would you prove that the Earth is, in fact, round?
 A: You can build a simple pendulum and observe how it rotates as the day progresses. You can then put a pendulum on a stick or something that you can rotate yourself in order to demonstrate that when you rotate the stick, the pendulum will continue to swing in the same direction. This shows that the direction of movement of the pendulum will change relative to its base only if its base is rotated. 
Pendulums can also be used to measure your latitude (its direction will change at different rates for different latitudes), and to measure the local value of g (the amount of time it takes to go through one cycle, or its period of oscillation, will vary with gravity).
A: The occurrence of noon (i.e. meridian passage of true Sun) isn't simultaneous for two observers situated along an east-west line. Hmmm...okay perhaps even simpler. Sunrise and sunset aren't simultaneous for those two observers.
A: Classically, the gravitational force experienced by a mass $m$ above the Earth is given by the familiar,
$$F=G\frac{Mm}{r^2}$$
where $M$ is the mass of the Earth. In other words, a mass will experience a force which continually decreases as it distances itself away from the Earth. Now suppose the Earth was a flat infinitely$^{\dagger}$ large plane in $\mathbb R^3$ which is infinitesimal, with mass density $\sigma$ (per unit area, not volume). The gravitational potential $\Phi$ satisfies the Poisson equation $\nabla^2 \Phi = 2\pi G \sigma \delta(z)$, assuming the plane is at $z=0$.

The solution is given by $\Phi(z)= 2\pi G \sigma |z|$. The gravitational force is $-\partial_z \Phi$, which is always pointing towards the plane. Another feature is that the gravitational force is constant with magnitude $2\pi G \sigma$. In other words, no matter how high one is above the plane, the same forced is experienced. To be more realistic, if the plane had some non-zero thickness, the force would still be constant, but whilst inside there would be a 'jump' as depicted:

Hence, to determine if the Earth is flat, one would simply have to conduct an experiment to see how the gravitational force scales as one increases altitude. One will find $F \sim r^{-2}$ approximately, as expected, confirming the Earth is round. Of course, for sufficiently small variations in $r$, one may be fooled into thinking $F$ is constant since the change is minute, but it is measurable.

$\dagger$ For convenience, it is taken to be infinitely large; the conclusions remain the same, but the force will of course be different, since it will be dependent on $x$ and $y$ as well.
A: I think the simplest way is to have two sticks of same size put both of them perpendicular to the surface of the earth in the mid day sunshine and the gape between them is to be few miles and exact time mesure the angle of elevation or mesure the size of the shadow so both will be differ!
By several exams in a sysmetic order we can find that the earth is round.
A: Simplest, you say? There are two that strike me as being simple to demonstrate. Luckily someone on the internet has already spent some time to help us here to make these easy to illustrate:
1. Shadows differ from place to place


Eratosthenes carried out this experiment to determine the circumference of the Earth, already assuming its spherical shape; incidentally, the proof of such being consequential of the procedure.
However, a demonstration can be achieved by a simple, local experiment (as opposed to having a party venture to a distant enough point):
Take a piece of card (A3, or so), attach two obelisks to the card by their bases and, with a light source, produce shadows - now, slowly bend the card so that it becomes convex (that is, the side with obelisks attached bulging out) and watch the effect.
2. You can see farther from higher


There are numerous other ways of demonstrating that the Earth is round, or curved, at least, from analysing the center of gravity to simply observing the other round objects that are visible in space; but I believe these illustrations to be the simplest to comprehend.
Images sourced from SmarterThanThat
A: There's a video on youtube of a island a few miles away such that when you see the island from an elevation, you can see further to its base than you can when you see the island from the shoreline ( a demonstration of answer 1 above). I think this is the simplest way given that now we have zoom ability, anyone can do this kind of experiment on a clear day from any shoreline viewing something a few miles away.
http://www.youtube.com/watch?v=bco_p4V7-QU&feature=related
A: The shadow of the Earth on the Moon during an eclipse and the way masts of ships are still visible when the hulls are out of sight are the classical reasons.
A: Another way is the triple-right triangle:


*

*You move in a straight line for a long enough distance

*Turn right 90° degrees, walk in that same direction for the same distance

*Turn again to the right 90° degrees and walk again the same distance


After this you'll end up at the starting point. This is not possible on a flat surface since you'd just be "drawing" an incomplete square.

Source: http://www.math.cornell.edu (add /~mec/tripleright.jpg to find the image)
A: If you're in the northern hemisphere, walk to the south and notice how Polaris sinks lower and lower each night until it disappears below the horizon. If he can handle more info point out how the stars before you are getting higher and the stars behind you are getting lower. Of course then you'll need to explain their east to west motions too.
A: Here's another way, which only works in the Southern hemisphere - at least that is the case if you use the Flat Earth Society's map of the world, where Antarctica is a ring of ice around the works, and the North Pole is at the centre.
When flying from Sydney to Santiago or Pretoria, you regularly get close enough to Antarctica to see it. If the earth were flat, that would mean a huge waste of fuel. Only on a spherical world does this kind of trajectory make sense. On a flat earth, the shortest distance from Sydney to Santiago is via the North Pole, Sydney to Pretoria takes you across the Himalayas.
Of course this depends on airlines trying to save fuel - if all of them are in cahoots with NASA and all the other conspirators who claim a spherical earth, then all bets are off...
A: If the person in question is from a temperate latitude, take them to the tropics to feel the heat of the noon sun, preferably trapped out on a sailboat without water. Point to the very high sun and make your point when they are the most miserable. Next, take them to a very high latitude. As they freeze and become exhausted at 3:00 AM while out walking the tundra, point out the low, non-setting (or non-rising) sun, and re-iterate your point in their heightened state of misery. Through suffering and a sense of pride, the object of your demonstration will now likely feel that they have "been there" and "seen it"  with "their own eyes". If convinced, that person will gladly proselytize the "truth" of aforementioned roundness of said planet, and will confront the heretics who do not believe.  
I think that there are no simple answers to provide "proof" of anything. "Proof" is relative, much in the way "truth" is relative. If simple means "without using science or technology" then you are without hope, as the receiver of the "proof" must accept the truth of the methodology.
Photos from space are photoshopped.
Ships at sea look below the horizon because Osirus/Neptune/Odin/Jesus/Bhaal does not wish man to see to infinity (which also proves that the heavenly bodies are not very far away).
Sticks in the dirt and shadows prove nothing unless you accept that other bodies are permanent, in orbital motion, and far away (at which point the person will already believe that the planet is round). 
Don't try to prove anything. You can't. Instead, "Demonstrate and educate", because all you can do is convince, not prove.
A: Sitting for a while by the seashore ought to make it clear the Earth isn't flat, even if you don't happen to see a ship go over the horizon. The edge of the discworld Earth would have to be just a few miles away, and there's no way that the entire, circular world would fit inside the circle that the ocean horizon seems to make.
Humans have not just known the Earth was spherical but actually have been measuring its radius for thousands of years. http://en.m.wikipedia.org/wiki/History_of_geodesy
A: Besides the going back in time option, you could just show your "numbskull cousin" a picture of the Earth taken from the moon like the one below. 

A: *

*Watch live ISS footage. Neither Hollywood nor any software can produce this raw footage at such level of realism.


*See planes flying live. See for instance how you can go from NY to London, then from London to Tokyo, and the from Tokyo to NY, always going eastwards. If still in doubt, try it yourself.
A: Showing a picture of the earth from space seems like the simplest way to prove that the earth is round, but first of all you would need to convince this tribesman that this photograph is taken from far up in the sky. This can be done by taking a video of the earth from takeoff to orbiting the earth. That way he can see the connection between the world around him and the blue dot on a black background as representing the same thing, the earth.
A: Seeing a plane disappear behind the horizon. If the earth wasn't round (i.e. flat) the plane would be visible for a longer time than if it was round, assuming it would be visible for a long time.
Of course, the earth could also have another, more exotic form, like a banana, kiwi, pear, or pine apple. Or maybe have a book form. So you have to make a distinction between two forms only: sphere-like or flat.
A: I am adding this answer because I feel like the other answer did not mention one aspect where we can very easily prove that the Earth is round (not flat), and it is doable with the naked eye, and no equipment is necessary.

I took this image myself, in a day of time where the moon is on the left (a little bit higher in the sky then the Sun), and the Sun is on the right, close to setting.

However, if you look at the image of the moon, it look like it is being lit from above a little bit, despite the Sun being much lower in the sky (relative to the Moon). It looks almost like the light is making a curved path across the sky as it propagates from the Sun to the Moon.
In reality this is caused by a lensing effect, and the curvature of Earth, and the curvature of the horizon (and the atmosphere). This is only possible if the Earth is round (not flat).
