Why does matter exist in 3 states (liquids, solid, gas)? Why does matter on the earth exist in three states? Why cannot all matter exist in only one state (i.e. solid/liquid/gas)?
 A: Not quite sure what you are asking, but I can explain the difference between the three common states of matter on a qualitative scale:
Solid: molecules form bonds with neighboring molecules, very little of these bonds are broken at any given time.
Liquid: molecules form bonds with neighboring molecules for most of the time, but there are enough energy for the bonds to break momentarily and be formed again with another molecule.
Gas: molecules almost never come close enough to one another to interact.
To form a bond, energy is released, to break a bond, energy is consumed, therefore, when the energy (represented by temperature) of some matter (such as water) is high, the state tends toward liquid and gas, and if enough energy is given in the form of heat, all bonds on individual molecules will break and release that molecule from the liquid or solid into gas.
The reason why there are multiple state of matter on Earth is because the Earth contains matters that melt/vaporize at different temperatures and the Earth has different temperatures at different places.
A: Because, in general, the state of matter reacts to heat in various ways. For example, at room temperature water is liquid. Remove the heat sufficiently from water and at some point (i.e. the freezing point) it will become solid (i.e. ice). Reheat the ice and it becomes liquid again. Add even more heat and it becomes a gas. If you keep adding more heat, eventually it will reach a plasma state.
Why a stone is solid and water is liquid (or they are both solid) at various temperatures occurs because different atoms have different reactions to other atoms and conditions within which they exist. It's a bit like asking why two different people think differently under the same conditions/circumstances – even though both are human, that which makes them different from each other is involved in why both do not react in the same manner.
A: The ultimate answer to a "why" physics question is "because".
Physics is about observing and measuring nature and then finding mathematical models that fit the measurements and predict new behaviors under different conditions. 
Because we have observed these four states of matter. we have formulated mathematical theories called thermodynamics and quantum mechanics that can describe the behavior of matter and predict its future behavior  in addition to describing a plethora of other  behaviors ( as how we can communicate on this board).
How this happens can be explained within the mathematical models.


*

*atoms and molecules are neutral  bound states of charges and mathematically there are spill over forces that create attractions and repulsions.

*the states are quantized, i.e. the bindings are not arbitrary and continuous but specific energy states are stable and others are not stable

*solids are when the energy states settle in lattice configurations and are at the lowest energy state.

*liquids happen when thermodynamic conditions, temperature and pressure, are such that  some of the bonds of the lattices are loosened,  and extra degrees of freedom appear.

*gasses appear when tempearture and pressure combinations loosen all the intramolecular energy level bindings and it behaves as an ideal gas

*plasma happens when the temperatures and pressures are such that the electrons are ejected from their orbitals and the gas becomes ions and electrons.
All these processes are described perfectly using quantum electrodynamics and thermodynamics as also described in the other answers.
That is the mathematical map of the nature we found ourselves in. (That's the way the cookie crumbles, That's the way the ball rolls, etc ) If there were only one phase, a different set of theories would describe them, not the ones that describe successfully our present world.
A: So far, no one has interpreted the question literally, so I will:
"Why cannot all matter [on earth] exist in only one state (i.e. solid/liquid/gas)?"
It could, but then we wouldn't be alive to observe it. Life is a nonequilibrium phenomenon. There are certainly places in the universe where all the matter is (more or less) in the same state, but they are cold (or very hot) and dead.
A: The Pauli Exclusion Principle -- no two Fermions can be in the same state.
That, and stuff is lazy: it "likes" being in a low energy state.
These, together with inter-particle forces, generate a few sets of statistical behavior for large numbers of particles.  The particular properties of the particles in question determine when and if they reach any of these states.
Below Plasma, the electrons energy levels are such that they can all "fit" in the potential well of various nuclei.  Laziness makes them go to this low energy state.  Pauli Exclusion makes them stack up on top of each other.
When you transition to Plasma, the electrons have so much energy that they leave empty states under them.  When an electron falls into such a hole (emitting a photon), it quickly picks up more energy from the other photons floating around and is kicked out again.  There is enough energy for electrons and nuclei to act independently.  You can see plasma in your every day life by lighting something on fire.
As the energy available falls, the electrons stack up on the available states around the nuclei.  In this state, the atoms (nuclei + electrons) or molecules (tightly electrically bound collections of atoms) move freely, and sometimes bounce off each other.  This is called a gas state of matter.
With the electrons mostly in their lowest energy state, they in a sense stop mattering.  A whole bunch of "degrees of freedom" go away.
Each atom/molecule still has lots of lower energy states it can go into, but much like above when an atom/molecule enters lower energy states, it often gets "kicked" up to a higher energy state by a faster moving other particle/photon hitting it.
Now, less energy.  Now the tightly-coupled atoms or molecules are moving slowly enough that the low energy states near them are full of other atoms or molecules.  They start coupling with said nearby atoms and molecules, and few have the energy to just "walk away".  If you compress it further, there aren't enough holes to fit the atoms/molecules in, so it pushes back.  If you decompress it, the higher volume opens up new states for atoms/molecules to slide into, so it sucks in energy.
This liquid state is relatively volume-constant, with lots of energy soaked up if you try to lower its density or increase it.  It requires lots of pressure to compress it.
However, there is still enough states for the individual atoms/molecules to move at a reasonably fast pace.
Now, less energy.  Now each molecule/atom is stuck in a trap.  Moving in any direction is confined by other particles nearby -- so your molecule pushes against the molecule in the direction it is going, and bounces back.  This makes the matter rigid.  Long-distance travel for molecules/atoms becomes exceedingly unlikely.  Increasing/decreasing volume usually becomes even more energetic.
In metals, the least-bound electrons in the molecules/atoms act somewhat fluid-like, in that they can flow around from one to another, as there are similar-energy states available nearby.
These are not the only states of matter, but rather the most common states of matter we interact with.  We can have Bose-Einstein condensate, quark-gluon plasma, electron degenerate matter (white dwarfs), etc etc etc.  Even at our typical temperatures and pressures triple-states can be reached where things behave like a mixture of the above.
You'll note that I talk about molecules and atoms and electrons above.  Molecules are bindings that can occur before "macroscopic" behavior occurs, but like anything else it can all become fuzzy at the limit.  Molecules can become large enough to be macroscopic (and arguably crystals are exactly that), and adding energy to the system can cause them to break apart before large-scale statistical changes in particle behavior can occur.
A: Basically the existence of different states of matter has to do with Inter-molecular forces, Temperature of its surroundings and itself and the Density of the substance.
This image below shows you how the transition between each states occur (called Phase transitions).

These transitions occur based on the change in temperature of the substance
Now if you compress (increase pressure) and reduce the temperature of gasses like $CO_2$ then it can exist in solid state which is generally called Dry Ice (5.18 bar, - 56.6°C)

But there are other exotic states of matter out there, like Plasma and Bose–Einstein condensate 
A: The premise is wrong. Not all materials exist in exactly three different states; this is just the simplest schema and is applicable for some simple molecular or ionic substances.
Let's picture what happens to a substance if you start at low temperature, and add ever more heat.
Solid
At very low temperatures, there is virtually no thermal motion that prevents the molecules sticking together. And they stick together because of various forces (the simplest: opposite-charged ions attract each other electrostatically). If you picture this with something like lots of small magnets, it's evident enough that you get a solid phase, i.e. a rigid structure where nothing moves.
Actually though:


*

*Helium won't freeze at any temperature: its ground state in the low-temperature limit at atmospheric pressure is a superfluid. The reason is that microscopically, matter does not behave like discrete magnets or something, but according to quantum mechanics.

*There is generally not just one solid state. In the magnet analogy, you can build completely different structures from the same components. Likewise, what we just call “ice” is actually just one possible crystal structure for solid water, more precisely called Ice Ih. There are quite a lot of other solid phases.


Liquid
Now, if you increase temperature, that's like thoroughly vibrating your magnet sculpture. Because these bonds aren't infinitely strong, some of them will release every once in a while, allowing the whole to deform without actually falling apart. This is something like a liquid state.
Actually though:


*

*Not all materials have a liquid phase (at least not at all pressures). For instance, solid CO2 (dry ice) sublimates at atmospheric pressure if you increase the temperature, i.e. it goes immediately into the gas state.

*Many materials have huge molecules, i.e. the size of the chemical structure approaches the size of the physical structure. Now, that chemical structure can also be shaken loose by heat, but this isn't called melting but decomposition then. For instance, plastics decompose at some point between 200°C and 350°C. Some melt before i.e. they have two states; some stay solid all the way, they basically just have one state (solid).
A decomposed material hasn't entered a new state of matter, it simply has ceased to be the original material.


*

*Furthermore, materials that aren't purely composed of one kind of molecule also generally don't have a simple fix melting point. There's a certain range in which two phases may coexist. (More generally, you can have all sorts of emulsions, dispersions, gels etc.)



Gaseous
Small and sturdy molecules or single atoms aren't so bothered by high temperatures though. They also don't have so strong forces between molecules. So, if you shake strongly enough, they simply start fizzing all around independently. That's a gas then.
Actually though:


*

*Even the most sturdy molecules won't survive if you make the temperature high enough. Even single atoms will at some point lose their hold on the electrons. This results in a further phase, a plasma.

*At high enough pressure – above a critical point, the gas phase won't really be distinguishable from the liquid one: you only have a supercritical fluid. (IMO this could still be labelled a gas, but it does have some properties which are more like a liquid.)



Now, the question why a particular material is in some particular state at some given temperature and pressure isn't easy to answer. You need statistical physics to predict the behaviour. The crucial quantities are energy and entropy. Basically, the random thermal motion tends to cause disorder (which is quantified by rising entropy). At any given temperature there's a corresponding amount of energy available to overcome the attractive force, and within that energy budget the system approaches the state with the highest entropy. A solid has little entropy, but if there's not much energy available this is the only feasible state. A liquid has higher entropy but requires some energy to temporarily unstick the molecules. A gas requires enough energy to keep the particles apart all the time, but is completely disordered and therefore has a lot of entropy.
But how much energy and entropy a given state has exactly varies a lot between materials, therefore you can't simply say solid-liquid-gas.
A: This is one of those funny questions where the cart gets put before the horse.  Matter doesn't "exist" in any state.  It simply does what it does, in the way it does it.  Humans, wishing to understand how different types of matter behave chose to create a system of three states.
This choice is the key: the reason "matter exists in 3 states" is because we chose to model it that way.  It would be trivial to declare "matter exists in 5 state" or "matter exists in 2 states."  We, in general, have chosen to treat 3 states, solid liquid and gas (plus plasma), as "fundamental" not because they're actually fundamental to physics, but because our choice of those divisions helps us predict how the materials will behave when they are interacted with.  For example, we find that the way a solid object, like a rock, behaves is fundamentally different from a liquid, like a stream of water, because for the kinds of things we worry about, its a useful distinction.  Getting hit in the face with a rock is typically a very different event than getting sprayed with water.
We do have rationales for why these states occur, based on the concept of intermolecular forces.  In a solid, molecules have very little freedom of movement because the intermolecular forces trap them.  Solid things have rigid behaviors.  In a liquid, molecules have enough freedom of movement to go anywhere in a volume, but the intermolecular forces still have a large effect on how they behave.  This mobility leads to traits we found important enough to categorize, such as fluidity.  In gases, molecules have so much freedom of movement that the intermolecular forces become more of a side note when it comes to predicting their behaviors.
What we have found is that, in many cases, the lines between these behaviors are rather sharp.  The transition between solid to liquid or liquid to gas tends to happen very close to a particular temperature.  Noe that I say close: the process of boiling or freezing is a statistical one, not an exact one.
For most of what we do, these two divisions, between solid and liquid and between liquid and gas, are effective enough at helping us understand the universe that we consider them "fundamental."  However, not everyone agrees.  High energy physicists consider the case where the thermal energy of a gas gets so high that it starts to strip its own electrons off, becoming nothing but a bunch of ions.  This material behaves differently enough from gas that they declared it a new "fundamental" type (for one thing, it's affected by magnetic fields!).
It has been found, that for many materials, its properties are well described by these categories, so we keep them!
On the other extreme, there's many cases where "solid" is not actually enough to capture the behaviors we care about.  In these cases, we adapt.  My favorite example is chocolate, because chocolate is a strange beast of a material.  You can melt it (solid to liquid), and the crystals of chocolate fat dissapear as you'd expect.  However, some crystal structures are more robust than others, requiring higher temperatures.  Likewise, the crystals form at different temperatures as you cool it.  This leads to some remarkable chemistry.  As it turns out, there are 6 "polymorphs" of the chocolate fat crystal, each with their own properties.  Of them, only Form V is good for chocolateering.  It's the crystal which has the charactaristic snappy crunchy feel we want from chocoloate.
Thus, when one tempers chocolate, one first raises the temperature to melt all crystals.  Then one reduces the temperature to cool it down and form crystals (the more the merrier).  All sorts of crystals form as the fats turn to solid, Form I through Form V (Form VI is different, and is associated with blooming).  After this, you raise the temperature to between 81.1F and 92.8F, which is the melting point of Form IV and the melting point of Form V respectively.  This lets all of the Form I through Form IV crystals melt, but keeps the Form V ones around.  Then, one pours the chocolate and let it cool, leaving only Form V crsytal structures.
Note that all that I talked about dealt with solids, crystal growth.  Through the entire process, the average layman would call that material 'liquid,' but I'm constantly freezing and melting things within that liquid state.  The simple concept of "liquid" just isn't enough.
A: To try to answer what I think is your underlying question, rather than the specific wording you use...
The electro-magnetic forces are only so strong. Let's say you have a box half-full of some molecule. Electro-magnetism keeps the individual atoms together (keeping the electrons bound to the nuclei) and it keeps the molecules themselves together (which, simplified, is actually the same as the previous case - the key is keeping the electrons bound to the nuclei again; it's just that the electrons are shared between two nuclei at a time to an extent). Finally, the molecules in the body can be held together by the same electro-magnetic forces to form solids or liquids.
When dealing with states of matter, we usually most frequently talk about heat and pressure. To simplify, I'm going to merge the two together - it's not very useful in practice, but let's just see where we get. We've already said that the individual molecules (let's pretend that all matter is made out of molecules for now) have some kind of attraction between themselves. These "bonds" have a certain potential energy - basically, a measure of how much energy you have to add to break the "bond". For example, a molecule of nitrogen holds together much more strongly than a molecule of oxygen, so you need more energy to break down nitrogen than you need to break down oxygen. One way to look at heat is as the average kinetic energy of the individual parts that make up matter, which is useful when thinking about the states of matter. The higher the heat, the higher the chance that any given "collision" will have enough energy to break that inter-molecular "bond" that determines the state.
Given the four basic states of matter, then:


*

*Solid - the bonds between the individual molecules are much stronger (require more energy to break) than the heat. The molecules thus form a relatively rigid structure where molecules  "stay in place".

*Liquid - the bonds between the individual molecules are strong enough to keep a surface. They are weak enough that the random heat effects keep breaking and reforming the bonds continuously, so the molecules are relatively free to move, rather than being kept in place in the rigid structure of a solid. They don't maintain a macroscopic shape, but their volume is pretty much constant. This works both ways - they don't expand like gases, and they barely compress at all.

*Gas - the bonds between the individual molecules are no longer strong enough to give any structure to the gas. The individual molecules barely interact with each other, and gases readily expand and compress because of that. If you compress a gas enough, you will get a liquid (and finally, a solid) - you're basically forcing the individual molecules to come close enough, and reinforcing the intra-molecular forces with external pressure.

*Plasma - the heat is so big that it not only disrupts any bonds between molecules, they also break apart the molecules and strip electrons from the individual atoms. Overall, plasma behaves in a similar way to a gas, with a few extra interesting properties.


All in all, it's a balance between all the forces acting on the constituents of matter. Imagine the wind-blown lottery machine, with a fan on the bottom and a bunch of balls. And just to make it a bit more real, imagine the balls are sticky. As you increase the air flow from the fan, you'll see (in a sequence):


*

*Balls sitting and wiggling on the ground - a "solid". The stickiness is enough to prevent movement.

*Balls jumping around and moving, but still mostly keeping to the ground - a "liquid". The stickiness is no longer enough to prevent movement, but it still holds the bulk together, in tandem with the pressure provided by gravity.

*Balls jumping all the way around the container, bouncing off the walls (and each other, but don't forget that molecules are absurdly tiny compared to the balls - an actual gas doesn't get all too many collisions) - a "gas". Neither the stickiness nor gravity are strong enough to restrict the movement of the balls anymore.

*The sticky surface is blown off the balls and freely moves all over the container - a "plasma".


The air flow from the fan is the "heat" analogue, and the strength of gravity provides us with the pressure. Increase gravity, and the balls will remain solid or liquid under higher air flows ("temperatures"). Increase stickiness ("electro-magnetic force" - in reality, different moleculess have different stickiness), and the balls will remain solid or liquid under higher air flows.
