How do I explain to a six year old why people on the other side of the Earth don't fall off? Today a friend's six year old sister asked me the question "why don't people on the other side of the earth fall off?". I tried to explain that the Earth is a huge sphere and there's a special force called "gravity" that tries to attract everything to the center of the Earth, but she doesn't seem to understand it. I also made some attempts using a globe, saying that "Up" and "Down" are all local perspective and people on the other side of the Earth feel they're on the top, but she still doesn't get it.
How can I explain the concept of gravity to a six year old in a simple and meaningful way?
 A: The misconception likely comes from a misunderstanding of "down".  Making a 2-D drawing of the earth with buildings, people, and trees might help.  For example, 
A: Wrap a ball (like a tennis ball) with a rubber band. Tell her to put her finger between the ball and the rubber band and try to move her finger away from the ball. Have her do this on all sides of the ball. Now explain to her how the rubber band is like gravity. 
A: Why don't you try to explain that every object attracts other objects, but that in order for it to be felt, at least one of the objects must be very big? The earth is a very big object, so it attracts things towards it. 
A: Point out the fact that a ball falls straight down, and not sidewards.  This is because it wants to go to the center of the Earth.
This is true everywhere.  What we think is down is towards the center of the Earth, and this is true whereever we go.
(and then have a look at the globe and see what direction towards the center is)
A: Rub a balloon with a cloth to induce a charge which will demonstrate "static cling", then attract small pieces of paper with the balloon.  Once they see that static electricity attracts the small pieces of paper to the bottom of the balloon, then you can start explaining to them about 'the force'.
A: Ask the child what "down" would mean to the people on the "other side" of the Earth.
A: Here are some ideas:
a) Try to make her understand the concept of "force" : tie two balls on an elastic band. Make her pull them apart so that she gets a feel  of force. Take an apple and drop it. Let her understand that if a force ( her hand) does not hold an object it is pulled by the earth, the way the elastic pulls the balls.
b) Then show her on a globe where you are. Show the vertical where the apple drops. You could then go to the force pulling towards the center, the way the elastic pulls along the line, and that it is the line that defines which way  the force pulls, both for the elastic and the earth.  Make the analogy that each point on the earth pulls along the line as if there is a band, towards the center.
good luck
A: Take a magnet and hold it vertically, sprinkle iron fillings on either side of the pole. Iron fillings on the bottom side will be hanging. Then the child will get a feel that, it is possible for things to stay without falling down. You need not explain the concept of gravity for the child now. The child will create the explanation for itself (might also become next newton by creating a new concept).
A: Tell him that down means "in towards the center of the Earth, no matter where you are."
A: After carefully considering the OP's situation, I believe the focus falls on the rhetorical process.  Following from this, my argumentative approach would be the following: stand firm about the fact that Earth is round, get the child to reconcile the inconsistencies.
The reason for this kind of approach is the child's behavior.  The following quote is a script that I have repeated over and over in physics tutoring:

I also made some attempts using a globe, saying that "Up" and "Down" are all local perspective and people on the other side of the earth feel they're on the top, but she still doesn't get it.

Here, you presented an argument.  But what followed after that argument?  There's no apparent commentary on the argument by the child.  I have no expectation that a kid would retort with consistent or event coherent counter-arguments, but the response seems to be missing altogether.
This is familiar to those of us with physics teaching experience.  Wrong answers are always easy to deal with, and almost universally constructive.  It's the lack of any model formation that blockades progress, and often leads them to switch majors to something non-technical because of the bad experience.
Consider the rhetoric to be like a chess game (with formally established rules for movement).  As an educated adult, you probably don't have trouble responding to any move the child makes.  If you do, ask another question here.
No rhetorical approach will be helpful 100% of the time.  Counter-examples aren't always helpful either, but they appeal to a very particular example of inconsistent logic.  If you ascribed to the bad logic before you see the counter-example, then it will accomplish it's purpose - demonstrating that you're wrong.  I think the best response I've seen here was the following image (posted as a comment):

(image originally from http://www.caloi.com.ar/caloidoscopio_new/byn/byn53.gif)
This is physics perfection.
The absurd illustration makes the viewer abandon a view.  You could reject that Earth is a sphere, or you could reject that gravity is always in the same direction.  I suppose the option remains that people are presently, at this moment, falling off the side of the planet.  I think that's what makes physics fun.  For each model we build, there is a story that goes along with it.  Most individual propositions can have a model built around it, changing everything else in the universe to accommodate.  But once you are forced to explain multiple facts simultaneously, then you are in the process of building physics.  Each model is a story.  If you can start enjoying the process of telling those models/stories, then you are on your way to grad school.
I say physics involves two things: rationalism and evidence.  Your model's consistency is dictated by reason, but which model applies is determined by evidence we get by the world around us (for which there is no mental shortcut).  Here are some stories that do have consistency:


*

*Earth is flat, gravity is always down.  The validity can more-or-less only be determined by measuring the Earth's shape.

*Earth is round, gravity is always the same direction.  A privileged "top of the Earth" location exists, and everything else slopes down.  You could verify this by observing a ball to roll off the edge of the world.


Maybe the best thing to do isn't to be correct, but show how you, yourself, enjoy being wrong.  If they emulate that behavior, they will be fantastic scientists.
A: Having my own 6-year-old and having successfully explained this, here's my advice from experience:


*

*Don't try to explain gravity as a mysterious force. It doesn't make sense to most adults (sad, but true! talk to non-physicists about it and you'll see), it won't make sense to a 6yo.
The reason this won't work is that it requires inference from general principles to specific applications, plus it requires advanced abstract thinking to even grasp the concept of invisible forces. Those are not skills a 6-year-old has at their fingertips. Most things they're figuring out right now is piecemeal and they won't start fitting their experiences to best-fit conscious models of reality for a few years yet.

*Do exploit 6-year-old's tendency to take descriptions of actions-that-happen at face value as simple piecemeal facts.

Stuff pulls other stuff to itself. When you have a lot of stuff, it pulls other things a lot. The bigger things pull the smaller things to them.

Them having previously understood the shape of the solar system and a loose grasp of the fact of orbits (not how they work—that's a different piece—just that planets and moons move in "circular" tracks around heavier things like the Sun and Earth) may be useful before embarking on these parts of the conversation. I'm not sure, but that was a thing my 6yo already had started to grasp at this point.
These conversations were also mixed in with our conversations about how Earth formed from debris, and how the pull was involved in making that happen, and how it made the pull more and more. So, I can't really separate out that background; it may also help/be necessary.

*Don't try to correct a 6-year-old's confusion about up and down being relative, but use it instead.

There's a lot of Earth under us, and it pulls us down when we jump. If we jumped off the side, it would pull us back sideways. If we fell off the bottom, it would pull us back up.

You can follow this up later with a Socratic dialogue about the relative nature of up and down, but don't muddy the waters with that immediately. That won't have any purchase until they accept the fact that Earth will pull you "back up" if you fall off.

*Build it up over a series of conversations. They won't get it the first time, or the tenth, but pieces of it will stick.

*Don't try to instill a grasp of the overall working model. If you can successfully give them some single, disconnected facts that they actually believe, putting them together will happen as they age and mature and get more exposure to this stuff.
All this is assuming a decently smart but not prodigious child, of course. (A 6-year-old prodigy can probably grasp a lay adult's model of gravity, but if that's who you're dealing with then you don't need to adjust your teaching.)
For some more context, this was also after my child's class started experimenting with magnets at school. I was inspired to attempt to explain gravity when my kid told me that trees didn't float off into space because the Earth was a giant magnet. (True! But not why trees don't float away.) Comparing gravity and magnetism might help, to give them an example of invisible pull that they can feel, but it might just confuse the subject a lot too since I had a lot of work (over multiple conversations) to convince my own that trees aren't sticking to the ground because of magnetism, even if the Earth is a giant magnet.
And, a final piece of advice that's incidental, but can help:


*

*Once you've had a few of these conversations, play Kerbal Space Program while they watch. (Again, this comes from experience. My kid loves to watch KSP.) Seeing a practical example of gravity at work in it natural environment will go a long way to cementing the previous conversations. It may sound like a sign-off joke, but seeing a system moving and being manipulated makes a huge difference to a young child's comprehension, because it is no longer abstract or requires building mental abstractions to grasp, like showing them a globe does.

A: There's a good chance you won't be able to make her understand how it works.
According to developmental psychologist Jean Piaget, children at her age tend to see themselves as the "center of the world", and are incapable of reasoning about it from any other viewpoint. The process is called egocentrism, and is the same reason why a kid gets confused when its mother calls its (the mother's) parents mother/father instead of grandmother/grandfather. 
It won't hurt trying to explain, and she may gain some insight, but don't get frustrated if she doesn't.
A: Explain that since the earth is round, and people do not fall down (as there is no absolute "down"), they fall towards the center of the earth.
A: What helped me to understand gravity is to model spacetime as a trampoline or a stretched bed sheet. Now put a melon on it, saying "This is earth which bends space." Now take a marble and say "This is you, also bending space, yet not as much". And no matter where you put the marble, it always goes "down" towards earth, as earth bends spacetime the most.
Yet what I think is important to convey that not only does stuff fall towards earth but that every mass has its own gravity field not entirely unlike a magnetic field.
You can go crazy with this experiment, saying the melon is the sun, a larger marble is the earth. Instead of letting the earth marble fall give it a sideway push and it will orbit the sun for a while.
You can even add a smaller marble as the moon and if you throw the earth and the moon marbles well enough, you will see the moon marble orbiting the earth marble while both orbit the melon sun.
There is a real nice example of this experiment on youtube.
A: This question is in fact a very challenging one and was first satisfactorily solved by Albert Einstein in the course of developing general theory of relativity from special relativity:
His first step toward a relativistic theory of gravitation was the proposition of the principle of equivalence. Equipped with this principle, he could explain some gravitational effects like gravitational frequency shift of light (or time dilation) and gravitational deflection of light:

but he couldn't account for the gravitational effects near gravitational sources like the Earth: he couldn't explain why the people on the opposite side of the Earth experience a gravitational pull in the opposite direction of the Earth. This effects are called Tidal effects:

He solved this problem by proposing a deep analogy between tidal forces and a property of surfaces called curvature. Basically, he proposed that space(time) can be represented by some curved geometrical objects called (pseudo-riemannian) manifolds. Thus, all you have to explain to him is this concept. I think the following picture, (although, with having the time dimension omitted, it's not precise at all) can help you/him a lot:

This pictures helps him to grasp the concept in a simple, geometrical and yet as scientifically precise as possible: all objects around the Earth fall toward the surface (center, of course) of the Earth. I think this explanation is in accordance with Einstein's quote everything should be as simple as possible, but not simpler.
To complete this answer, I explain (very) briefly the remaining steps to general relativity:
After this point, the only remaining step toward his theory was finding a relationship between this curvature and presence of gravity sources like matter and radiation. He arrived at his famous equation:
$$\mathbf{G}=\frac{8\pi G}{c^4}\mathbf{T}$$.
in which the quantity $\mathbf{G}$ measures curvature of space(time) and the quantity $\mathbf{T}$ measures matter content.
Thus, a complete solution to the question why objects placed at different locations around the Earth experience forces in different directions (all toward the center of the Earth) necessarily needs a general-relativistic argument.
