If the gravity at the center of the Earth is zero, why are heavy elements like iron there? If gravity is zero at the center of the earth, why is there a core of heavy elements, such as iron? 
Alternate question for the opposite hypothesis:
If gravity is greatest at the center of the earth, as classical education tells us, why is the core not dominated by the heaviest elements (elements heavier than iron)?
I am a person reasonably familiar with technical terms, but I am not a physicist so I will appreciate answers that don't rely on equations. I am 70 years old and I want to explain it to my mother who is equally curious. 
 A: Forget about force. Force is a bit much irrelevant here. The answer to this question lies in energy, thermodynamics, pressure, temperature, chemistry, and stellar physics.
Potential energy and force go hand in hand. The gravitational force at some point inside the Earth is the rate at which gravitational potential energy changes with respect to distance. Force is the gradient of energy. Gravitational potential energy is at it's lowest at the center of the Earth.
This is where thermodynamics comes into play. The principle of minimum total potential energy is a consequence of the second law of thermodynamics. If a system is not in its minimum potential energy state and there's a pathway to that state, the system will try to follow that pathway. A planet with iron and nickel (and other dense elements) equally mixed with lighter elements is not the minimum potential energy condition. To minimize total potential energy, the iron, nickel, and other dense elements should be at the center of a planet, with lighter elements outside the core.
A pathway has to exist to that minimum potential energy state, and this is where pressure, temperature, and chemistry come into play. These are what create the conditions that allow the second law of thermodynamics to differentiate a planet. As a counterexample, uranium is rather dense, but yet uranium is depleted in the Earth's core, slightly depleted in the Earth's mantle, and strongly enhanced in the Earth's crust. Chemistry is important!
Uranium is fairly reactive chemically. It has a strong affinity to combine with other elements. Uranium is a lithophile ("rock-loving") element per the Goldschmidt classification of elements. In fact, uranium is an "incompatible element", which explains the relative abundance of uranium in the Earth's crust.
Nickel, cobalt, manganese, and molybdenum, along with the most extremely rare and precious metals such as gold, iridium, osmium, palladium, platinum, rhenium, rhodium and ruthenium, are rather inert chemically, but they do dissolve readily in molten iron. These (along with iron itself) are the siderophile (iron-loving) elements. In fact, iron is not near as siderophilic as the precious metals. It rusts (making iron is a bit lithophilic) and it readily combines with sulfur (making iron a bit chalcophilic).
This is where pressure and temperature come into play. Pressure and temperature are extremely high inside the Earth. High pressure and high temperature force iron to relinquish its bonds with other compounds. So now we have pure iron and nickel, plus trace amounts of precious metals, and thermodynamics wants very much to have those dense elements settle towards the center. The conditions are now right for that to happen, and that's exactly what happened shortly after the Earth formed.
Finally, there's stellar physics. The Earth would have a tiny little core of rare but dense elements if iron and nickel were as rare as gold and platinum. That's not the case. Iron and nickel are surprisingly abundant elements in the universe. There's a general tendency for heavier elements to be less abundant. Iron (and to a lesser extent, nickel) are two exceptions to this rule; see the graph below. Iron and nickel are where the alpha process in stellar physics stops. Everything heavier than iron requires exotic processes such as the s-process or those that occur in a supernova to create them. Moreover, supernova, particularly type Ia supernovae, are prolific producers of iron. Despite their relatively heavy masses, iron and nickel are quite abundant elements in our aging universe.

(source: virginia.edu)
A: Here's an interesting thought experiment.
Imagine you have an elevator shaft to the centre of the Earth which, for some strange reason, doesn't affect the gravitational field of the Earth and doesn't flood with magma.
OK, now at the Earth's surface get a bottle, half full with oil and half full with water.   The water is denser than the oil, so the force of gravity on the water is greater than the force of gravity on the oil... so the water sinks to the bottom and the oil floats on the top.
Now, head down your elevator shaft.  Is the gravity weaker or stronger here?  Well, for our bottle of oil it doesn't really matter.  Whatever the gravity is, it still produces a greater force on the water than it does the oil, so the water will always sink.
In terms of materials floating or sinking relative to other materials, it doesn't matter where the gravity is strong of weak, what matters is only the direction of the gravity.
So why isn't the Earth a big sphere of materials layered by density?  Well... largely it is.  Iron (7,870 kg/m^3) is denser than magma (~2,500 kg/m^3) is denser than water (1000 kg/m^3) is denser than nitrogen (~1 kg/m^3)... and that's the order you generally find them in.  
What about the exceptions? Why is there gold (19,300 kg/m^3) and iron in the Earth's crust... I suggest David Hammen's post.
A: There are two different quantities here to distinguish: the gravitational force and the gravitational well. At the center of the Earth, the gravitational force is zero, but the gravitational well is at its deepest. The heavy elements tend to migrate to the lowest point in the gravitational well, so they are at the center, even though the force is zero there.
If I drop a ball here on the surface of Earth, it will accelerate downwards at about $10\, \mathrm{m/ s^2}$ This is because the gravitational force pulls it down. Gravitational force pulls things toward the center of the Earth. As you go higher and higher up, the gravitational force gets weaker. If you go up a tall building, the gravitational force goes down by a few thousandths of a percent, but if you go way out into space, say as far as the moon, it gets much weaker, eventually getting so weak you can barely notice it any more.
As you go down into the Earth, the gravitational force gets stronger because you are getting closer to the heavy stuff at the Earth's center. However, if you go down thousands of miles (much further than we have the technology to go today), the gravitational force will start getting weaker because most of the Earth's mass is above you now and is no longer pulling you down towards the center. So gravitational force maxes out part way down towards the center, then starts fading away. At the very center, the gravitational force is zero because there's equal mass pulling on you from all sides, and it all cancels. If you built a room there, you could float around freely. That's what it means to say that gravity is zero at the center of Earth.
However, the gravitational well is a different story. This is about how much energy it would take to escape Earth. If you're on the surface of Earth, this is about 60 million Joules per kilogram. As you go up, it gets smaller and smaller, and if you go out very far, it effectively falls to zero once you're far enough away that Earth's gravitational pull is negligible.
As you go down deeper into the Earth, you get deeper and deeper into the gravitational well. Even when you're deep in the Earth and the gravitational pull is not very strong, going further down still moves you deeper into the Earth's gravitational well.
The gravitational force and the gravitational well are related to each other. The force is how fast the well gets deeper. When you get deep in the Earth, but not quite at the center, the gravitational force is small. That means that moving further down puts you deeper into the gravitational well, but only gradually. The slope of the well is shallow there, but still getting deeper.
Roughly speaking, the elements in a planet like Earth will try to minimize their energy. They do this by getting as deep into the gravitational well as they can because the deeper they go into the well, the lower their energy. The deep parts of the well do fill up, though, because not everything can fit down at the very center. The energy is minimized by putting the heavy stuff, like iron, down at the center, and the lighter stuff higher up.
This is far from a perfect description of Earth because it's what happens at equilibrium and at zero temperature, and that's not Earth, but it's a decent rough approximation of what happens in Earth.
So your answer is that gravitational force is zero at the center, but gravitational energy is lowest there, and heavy things go to where gravitational energy is lowest, so that's why the center of Earth is mostly the heavy stuff.
A: I will try to make a very approximate answer for your mother (as requested), assuming
the Earth spherical, and several other approximations. I am no expert
in geophysics, or stellar physics. and if you want details or greater
accuracy, I suggest you look at other answers, such as that of David
Hammen and others.
About gravity
First regarding gravity. Is there gravity at the center of the Earth,
and if not, why should anything be attracted there?
A basic exercise when studying gravity is to
compute the force gravity inside an empty spherical shell of matter
(like the rubber of a basket ball). The answer is: there is no gravity
produced by the spherical shell inside the shell , though there is
gravity outside produced by the shell.
If you now consider a shere filled with matter, will a 6371 km radius
(like the earth), and a point at 5000 km from the center, you can
decompose it into a full sphere of 5000 km radius and a spherical
shell around it with 1371 km thickness. The spherical shell causes no
gravity, hence, all the gravity there is to be observed is that
produced by the sphere of 5000 km radius.
This is actually true for any radius, so that, at the center of Earth,
i.e. with a radius 0 km, there is nothing left to produce gravity
since all the matter is in the "shell".
But that does not matter too much since, there is some gravity towards
the center as soon as you get at some distance from the center,
however weak when close to the center, so that with time, havier
matter will tend to sink to the bottom, i.e. to the center.
Then there is the issue of what is heavier.
What is Earth made off
Original matter in the universe (not going back to the Big Bang
though) is composed of mostly very light element, mostly hydrogene.
Stars form by accretion of this matter under gravitational forces, and
start fusing it (nuclear reaction) into heavier elements, and produce
energy we perceive (partly) as light. They tend to produce a lot of
elements like iron (and others that around the "middle" of the table
of elements, becauses these have the most stable atomic nucleus from
which little energy can be extracted, so that stars die (in various
ways) when they have transformed their matter into such
elements. Final explosion of some stars (supernovae) produce heavier
elements, but not in such great quantity.  This (very grossly) explains
why iron (and some other elements) tend to be available in greater
quantity.
Why is matter not stratified by density.
Again I am no expert, by there is a variety of phenomena that are at
work. Here are two examples.
Indeed, since at least part of the planet is somewhat fluid, one could
expect that te heavy components would sink. But there is much heat
produced inside the planet, due inparticular to radioactivity, and
this heat produces convection (and thus continental drift). Convection
means motion, moving matter around. That is more a dynamic aspect.
Another phenomenon is that chemical elements are seldom pure. They
combine physically or chemically to make composites that have
different physical properties. A compound formed of a heavy element
and a light one can be fairly light and float the heavy component
towards the surface of the planet, the lighter part playing the role
of a buoy. So, though uranium is much heavier than iron, uranium
composites with lighter elements can be found on the surface of the
planet, or very close to it. The phenomenon depends much on the
ability of the different kinds heavier elements to combine with
lighter ones.
You must also take into account that Earth took a long time to form and importance of different phenomena may have changed over the course of its formation.
A: Take a glass of water and two small balls, same size, one of iron and one of aluminum. Both will reach the bottom finally, but because of buoyancy the iron will settle first.

The Earth was discovered to have a solid inner core distinct from its liquid outer core in 1936,

.....

It is believed to consist primarily of an iron–nickel alloy and to be approximately the same temperature as the surface of the Sun: approximately 5700 K (5400 °C).

....

The Earth's inner core is thought to be slowly growing  the liquid outer core at the boundary with the inner core cools and solidifies due to the gradual cooling of the Earth's interior (about 100 degrees Celsius per billion years). Many scientists had initially expected that, because the solid inner core was originally formed by a gradual cooling of molten material, and continues to grow as a result of that same process, the inner core would be found to be homogeneous. It was even suggested that Earth's inner core might be a single crystal of iron. However, this prediction was disproved by observations indicating that in fact there is a degree of disorder within the inner core. Seismologists have found that the inner core is not completely uniform, but instead contains large-scale structures such that seismic waves pass more rapidly through some parts of the inner core than through others. In addition, the properties of the inner core's surface vary from place to place across distances as small as 1 km. This variation is surprising, since lateral temperature variations along the inner-core boundary are known to be extremely small (this conclusion is confidently constrained by magnetic field observations). Recent discoveries suggest that the solid inner core itself is composed of layers, separated by a transition zone about 250 to 400 km thick. If the inner core grows by small frozen sediments falling onto its surface, then some liquid can also be trapped in the pore spaces and some of this residual fluid may still persist to some small degree in much of its interior.

....

The Earth's inner core is a ball of solid iron about the size of our moon. This ball is surrounded by a highly dynamic outer core of a liquid iron-nickel alloy (and some other, lighter elements), a highly viscous mantle and a solid crust that forms the surface where we live.
Over billions of years, the Earth has cooled from the inside out causing the molten iron core to partly freeze and solidify. The inner core has subsequently been growing at the rate of around 1mm a year as iron crystals freeze and form a solid mass.
The heat given off as the core cools flows from the core to the mantle to the Earth's crust through a process known as convection. Like a pan of water boiling on a stove, convection currents move warm mantle to the surface and send cool mantle back to the core. This escaping heat powers the geodynamo and coupled with the spinning of the Earth generates the magnetic field.

So from this we see that the solid inner core slowly built up from the liquid outer core. It is in the outer core that the compositions differentiate the heavier elements precipitating from the liquid in the gravitational field , which arises from the inner core.

Extrapolating from observations of the cooling of the inner core, it is estimated that the current solid inner core formed approximately 2 to 4 billion years ago from what was originally an entirely molten core. If true, this would mean that the Earth's solid inner core is not a primordial feature that was present during the planet's formation, but a feature younger than the Earth (the Earth is about 4.5 billion years old).

Let us look then at the period when the inner and outer core were liquid. The closer to the center of the gravitational field the less gravitational force, but still the volume to mass* would play the same role in the liquid, concentrating the heavier to the center, forming the first seeds for the core as the system cooled.

why is the core not dominated by the heaviest elements (elements heavier than iron)?

Now the reason  the core is iron/nickel is due to the binding energy curve of the elements.


Binding energy per nucleon of common isotopes
The buildup of heavier elements in the nuclear fusion processes in stars is limited to elements below iron, since the fusion of iron would subtract energy rather than provide it. Iron-56 is abundant in stellar processes, and with a binding energy per nucleon of 8.8 MeV, it is the third most tightly bound of the nuclides. Its average binding energy per nucleon is exceeded only by 58Fe and 62Ni, the nickel isotope being the most tightly bound of the nuclides.

That is where fusion stops being energetically favorable.  In the Big Bang model  where a primordial soup ends up in the creations by fusion of nuclei, the model stops at the top of the curve.
Nuclear synthesis for heavy elements proceeds in supernovae explosions:

Elements above iron in the periodic table cannot be formed in the normal nuclear fusion processes in stars. Up to iron, fusion yields energy and thus can proceed. But since the "iron group" is at the peak of the binding energy curve, fusion of elements above iron dramatically absorbs energy. (The nuclide 62Ni is the most tightly bound nuclide, but it is not nearly so abundant as 56Fe in the stellar cores, so astrophysical discussion generally centers on the iron.) Actually, 52Fe can capture a 4He to produce 56Ni but that is the last step in the helium capture chain.
Given a neutron flux in a massive star, heavier isotopes can be produced by neutron capture. ...

In conclusion:

The layers containing the heavy elements may be blown off by the supernova explosion, and provide the raw material of heavy elements in the distant hydrogen clouds which condense to form new stars.

Because the heavier elements are much rarer  and coming from a secondary step as an explosion of a star the specific conditions of the formation of our star, the sun, and the creations of the planets around it show that earth, has heavier than iron elements accumulated at a second level to the original matter that coalesced to its core. The abundances are very small

the heaviest naturally radioactive elements, thorium and uranium, make up 8.5 parts per million and 1.7 parts per million, respectively. Some of the rarest elements are also the densest; these are the platinum group metals, including osmium at 50 parts per trillion, platinum at 400 parts per trillion and iridium at 50 parts per trillion.

and would not be detectable with the seismographic methods that study the inner and outer core.
             -----


*

*volume to mass for all elements can be seen here
A: Newton's law states that the center of spherical shell feels zero gravity.  So the very miniscule (actually the very point) center of the earth feels zero gravity (from the earth itself).  Think of it this way, every direction you look there is the same mass pulling radially away - the force of gravity all balances to zero.  Now move 100 miles in any direction from the center.  You now have a hundred miles of matter no longer in the shell exerting an unbalanced pull - gravity takes effect and separation of stuff starts to take place with denser material falling inward and lighter material floating upwards. The further you move from the center the higher the unbalanced force of gravity becomes and the faster the separation takes place.  Note that having zero gravitational pull at the center does not mean zero pressure.  The pressure from all of the unbalanced forces adds up despite the variations in gravitational attraction.  So the center of the earth feels no gravitaional pull, but does feels the greatest pressure, all due to spherical symmetry.
A: I think a simple answer is the boyant force is mostly responsible for heavy elements to sink to the center of the earth. For instance an aircraft carrier floats on water because the ship has a lot of space inside of it. Thus if you filled this space with water and then weighed it, you would find that the weight of the ship without the water is less then the amount of water it displaces or water weight. This makes it lighter than the water and it floats. Heavy elements have more atoms in a given amount of space than lighter elements so heavy elements sink while lighter ones float on top and so on.
An easy way of thinking about the center of the earth is that if you found yourself there, every direction from the center is up and the force would be the same from all directions so they would cancel out leaving you weightless.
A: I'm just 14, and I will try to answer the question based upon my understanding.
First of all, gravity, being a force and thus a vector, would cancel out in the core, as it not only depends upon the magnitude of the relative force but also the direction of it i.e. a vector going upward would cancel out with a vector going downward, and so on. But.....
If we were to carve a shell for ourselves at the center of the earth (Refer to the shell theorem) we would experience weightlessness inside the shell until we are in it. That would be experiencing zero gravity. According to your question, if the core were made of heavier elements, it would only affect the gravitational force we experience outside that shell.
So, it wouldn't matter if the core was made up of iron or tungsten. The core is made up of what it is and that's nature. You must be familiar with the history of the earth, how it was formed. Gravity wouldn't be in any effect with what our core was made up of.
But the real problem would be of the magnetic field. Iron is a superb magnet (when magnetized or found as a magnet). It was and is the sole supporter of our magnetic field. I don't know about many other elements, but a heavier element would certainly not be able to sustain our magnetic field. If it could, it would either be too strong or too weak to hold "cosmic radiation" from the sun. If too weak, the radiations would decimate us. If too strong, the same would happen.
A fundamental law of physics stated by Newton is that all particles attract each other, however it is so small (The Gravitational Constant) that we can only see the force of gravity for heavenly bodies (the planets and stars, and so on). So at the core, we would experience gravity, but not in the shell we would create, where the shell theorem would apply.
So, in short, nature has made our core and we cannot change it. We have never experienced (and I hope we never do) a change in the composition of the core. As for the question, I believe there wouldn't be any effect on the gravity at the core if the the elements composing it were different. But it could certainly make the gravitational pull we experience different. It could even render our planet inhabitable.
Hope this helps.
