Why does a fluid push upward on a body fully or partially submerged in it? Now you might think that I'm asking about the buoyant force. And you'd be correct, partially. You see, I understand why the net force on a body submerged in a fluid is upwards. But I want to know: why does a fluid push the body upward in the first place?
It's easy to understand why it pushes downward on the top of the body (because the weight of the water above it pushes down). I also understand the sideway forces. Actually, there's an easy way to prove the downward and sideward forces. Consider a container filled with some fluid. If you were to drill a hole in the container(below the water level) then no matter wherever you drilled the hole (on either side or at the bottom) the fluid would flow out. BUT let's say the container was enclosed and you drilled a hole on top of a container. This time the fluid wouldn't flow out even if the container was absolutely filled with water.
And that's essentially my question. How does the fluid exert a force upward on a body submerged in it? (NOTE: I know this might seem basic to some of you. But I have been trying to make sense of this for SOOOOO LOONG...so please don't close this down)
 A: In the left diagram the piston is held with the spring at natural length. It's then released and settles down to the diagram on the right.
The spring is then under compression and shows that there is a upward force on the piston.  This upward force is equal to the weight of a section of liquid with a height equal to the difference in levels between the piston and the red line on the right hand diagram.

So when an object is submerged in water, there is an upward force due to the pressure at the bottom of the object - it's $h\rho g$ where $h$ is the depth to the bottom of the object ($h_2$) and $\rho$ is density.
The pressure at the top is less, $h\rho g$ again but this time with a lower value of $h$, say $h_1$.  The resultant upward force is $A\times (h_2 - h_1)\rho g$, where $A$ is the area.
For a cuboid object, since volume $V = A\times (h_2 - h_1)$ we can get to Archimedes principle $$F = \rho Vg$$ 'Upthrust is equal to the weight of the fluid displaced'.
A: The fluid does not really exert an upward force on a body.  It exerts a force everywhere on the body normal to its surface.

[Adapted from Hyperphysics]
In a gravitational field, the pressure increases with depth, so those normal forces which would otherwise cancel, end up summing to an upward force vector. If the tank were placed on, say, a centrifuge, where the local acceleration pointed outward and caused a sideways pressure gradient, the object would likewise "float" sideways.
A: 
Consider a container filled with some fluid. If you were to drill a hole in the container(below the water level) than no matter wherever you drilled the hole(on either side or at the bottom) the fluid would flow out. BUT let's say the container was enclosed and you drilled a hole on top of a container. This time the fluid wouldn't flow out even if the container was absolutely filled with water.

Let me use the same line of reasoning to answer your question.
Take an empty bottle and punch a small hole in its bottom. Place it on the water surface. There will be no water shooting out of the hole.

Now press the bottle into the water. You will see that the water now shoots out of the hole. And, this is the origin of the upward push you feel while pressing the bottle into the water.

Note that even if the bottle is just floating on the water surface, it is pressing against the water ever so slightly due to gravity. This causes the water to exert a force (equal to the volume displaced by the bottle), and this is why the bottle floats.
A: On a macro level (aka how to understand how buoyant force works) - see trula's answer.
On a micro level - in solids, the normal force is basically just charges being repelled electromagnetically as they come close to each other. In liquids, molecules are fairly close to each other so one might think of it similar to sand. If you put something on sand, it might give way or wiggle around a bit but in general, while each of the grains can't hold the whole thing you put on it, they as a whole can.
Finally, in gases, molecules are constantly flying in all directions. If you take a sheet of paper and let it fall from some height, then poke it with something from below often enough, it could stay afloat pretty much indefinitely. This is how buoyancy works at a micro level - plenty of these tiny little pokes push whatever's submerged upwards. You could think of liquids similarly, just note that instead of poking the submerged object and flying away who knows where the liquid molecules stay close together and wobble, more like. Similar to a massage chair maybe? Except that it could give way entirely if the submerged object is dense enough. I don't think there's a 100% analogy - liquids are liquids - but hope all these analogies help you to gain some insight.
A: First consider holding an object between to hands, if the lower hand pushes harder the object will go up if the user pushes harder the object goes down.
in water pressure increases proportional to depth. $p=h*\rho*g$ the force is pressure times area $F=p*A$. Take a cuboid of height h1 and upper and lower area A
the pressure on the lower area is pressure at the higher area $ +h* \rho*g$ if the difference is higher than the weight of the cuboid it will go up, otherwise down.
Other thought: take any volume of water, it has some weight, wich makes a downward force mg, but it does not move down, so there must be a force upward wich is also mg.  now put the same volume of same other object in the same place, of cause the force on it will be the same namely mg where m ist the mass of water it replaces.
A: Gravity Sorts
Gravity is a kind of mindless sorting machine.  It's not a perfect sorting machine, but when you have objects that can slide past each other easily, it does a pretty good job.  The way it sorts is by pulling on objects based on their density.  The densest objects collect at the bottom.  The lightest objects "float" to the top.
Your question is very much like asking this: "If I hold a brick in the air, and then I let go of the brick, why doesn't the air under the brick cause it to levitate?"  This might seem like a silly question, but it contains some important insights.  First, the brick and the water scenarios are different, because air is highly compressible, whereas, water is mostly incompressible.  But they are similar because both air and water are very slippery.  It's extremely easy for air and water (and fluids in general) to move around in response to forces.  So when you let go of the brick, gravity pulls it down.  But it's also pulling on the air underneath the brick!  So why does the brick "win"?  Well, it's because the brick has a higher mass per volume, so when gravity pulls it down, the brick is able to push aside the air.
Bricks Float
Now, the same story also works for water.  Drop a brick in water, and it will fall to the bottom.  Or does it?  That depends on the brick, doesn't it?  If the brick is made of plastic and filled with air, then it doesn't fall to the bottom, does it?  Because the walls of the brick trap the air in a fixed volume, and because they are solid and don't move around much, the average density of the hollow plastic brick can be much less than water, causing the brick to float.
Now, we can say that the water is pushing up on the brick, but a more useful statement is that gravity is pulling down on the water.  When gravity "weighs" the volume of water taken up by the brick, and compares it to the mass of the brick, it says: "Well obviously, the water weighs more", so it pulls the water down more strongly.  And thus, the brick is left high and dry (mostly).
Bottles are not Fountains
If you have an open container filled with a fluid (not a superfluid!), the fluid doesn't spontaneously leak out the top (unless it's under pressure, but let's assume STP).  Again, we should not be surprised by this, because fluid isn't really trying to push up.  In both cases, fluid is being pulled down by the earth's gravity.  It's just that if you submerge something in the fluid that is less dense than the fluid, then the fluid will get pulled down harder than the object, and result is that the object gets pushed upwards.  If nothing is competing with the fluid for space, then the fluid doesn't exert any buoyant forces.  Put something into the fluid, and it will most certainly flow upwards out of the container.
Boats are Lighter than Water
It might seem counter-intuitive that an aircraft carrier or a petroleum supertanker is "lighter than water", but the only way they can float is if the average density of the ship is less than water.  Since most of a ship is hollow and filled with air, it doesn't matter that we build them with very dense, thick steel hulls.  The air lowers the average density enough to let the boat float.  Of course, a boat is a complicated kind of brick, because if it's working properly, much of the mass is above the waterline.  The boat's entire mass contributes to the "effective density", but only the volume below the waterline contributes.  And that's how we can tell how low the boat will float: it will sink until the volume below the water causes the average density to match water.
A: John Hunter's answer says why water flows out the bottom or side. This is why it doesn't flow out the top.
If you drill a hole in the side, there is water above the hole. The weight of this water pushes water out the hole. This continues until water reaches the level of the hole. At that point, the weight of the water above has dropped to $0$.
If you drill a hole in the top, the weight of the water above is already $0$.
A: 
BUT let's say the container was enclosed and you drilled a hole on top of a container.

That's not quite true. Suppose you were to take a bottle with you while deep sea diving, fill it at a deep depth, seal it, and then take it to the surface. If you then poke a hole in the top, there will be some water coming out of the hole.
There won't be much water coming out, because water isn't very compressible. Compressibility is how much the volume changes from being put under pressure, so low compressibility means that the volume changes little from a lot of pressure, but it also means that pressure changes a lot from a small change in volume. Once enough water comes out of the top so that the rest of the water reaches sea level pressure, the flow will stop.
With a hole in the side, the flow will continue until the water level reaches the hole, so it's more noticeable. If you have a hole in the bottom of a boat, water will shoot up from the hole because the surface of the water is above the hole. This phenomenon is often exaggerated to have the height of the fountain go above the surface of the water, such as here, which violates hydrodynamics.
A: This is called Pascal's law.
See also:

Due to the fundamental nature of fluids, a fluid cannot remain at rest under the presence of a shear stress. However, fluids can exert pressure normal to any contacting surface. If a point in the fluid is thought of as an infinitesimally small cube, then it follows from the principles of equilibrium that the pressure on every side of this unit of fluid must be equal. If this were not the case, the fluid would move in the direction of the resulting force. Thus, the pressure on a fluid at rest is isotropic; i.e., it acts with equal magnitude in all directions. This characteristic allows fluids to transmit force through the length of pipes or tubes; i.e., a force applied to a fluid in a pipe is transmitted, via the fluid, to the other end of the pipe. This principle was first formulated, in a slightly extended form, by Blaise Pascal, and is now called Pascal's law.

A: 
BUT let's say the container was enclosed and you drilled a hole on top
of a container. This time the fluid wouldn't flow out even if the
container was absolutely filled with water.

But, if you had a boat (which is a kind of an empty, open top container), and drilled a hole in it's bottom... You'd get the piston situation in John Hunter's answer - if the piston had a small hole in it.
P.S. You can take the analogy even further if you want. Expand the piston and make it concave (so that it is more like a boat, and so that it doesn't have to rely on the walls of the tube to keep the water out). Then expand the tube itself, and reshape it, so that it is more like a watering can. Then get rid of the spout, and expand it further - into large barell, a swimming pool, a lake...
A: 
It's easy to understand why it pushes downward on the top of the body
(because the weight of the water above it pushes down). I also
understand the sideway forces.

I don't think that it is possible to understand buoyancy without the help of the molecular structure of matter. If the molecules were a static stuff as a fine powder, there was no reason for any upward force.
But molecules have kinetic energy. They are moving and colliding with its neighbors, and with any surface immersed in the fluid.
In the absence of gravity, the pressure resulting from that collisions would be equal for any surface portion of the immersed object. But the acceleration of gravity makes the higher ones have more potential energy, while the lower ones have more kinetic energy.
So, the collisions at the bottom part of the object result in greater pressure than in the upper part, what is the source of the buoyancy force.
A: It's only a net upward force if the body can float, otherwise it's equal force from all sides. Imagine a lead weight sinking through water, there was an upward force acting on it as first began to sink, but those were overcome by the increased density of the weight+gravity, once it's submerged the fluid exerts even pressure on all sides.
