Why is it possible to drive a nail into a piece of wood with a hammer, but it is not possible to push a nail in by hand? It is possible to drive a nail into a piece of wood with a hammer, but it is not possible to push a nail in by hand.  Why is this so?
 A: There's many different things at work here.
First, there's the issue of acceleration. Hammers are very hard and solid, so when you hit the nail head with the hammer, the energy and force of the blow is delivered at almost an instant. Hands, on the other hand, are rather soft, and will spread out the same amount of energy and acceleration over a longer time period, resulting in a lower force applied on the piece of wood. Different woods have different resistance to pressure, so it's still rather easy to push a nail through a sheet of balsa wood, for example, while it's much harder to push it through a sheet of oak.
Second, the hammerhead actually accumulates a lot of energy in the duration of the swing, stored as kinetic energy in the head. That's why hammerheads are heavy (and the higher force you need, the heavier the head) - it allows you to store more kinetic energy with the same velocity of the head. The maximum velocity your muscles are capable of is a lot more limited than the amount of energy they can deliver, when considering something as tiny as a nail.
Third, hammers work as an additional lever, allowing you to deliver more force as a trade-off with time. This works in tandem with the second point - a longer swing can give you more impact force. This also helps the hammerhead reach higher speeds than you would have while holding the head directly, as opposed to holding the shaft.
Fourth, you're simply not going to hit as hard with your bare fist. Your body has built-in safety mechanisms that try rather hard to prevent injury, and you can hurt yourself quite a bit by hitting a nail head-on. Note that it's quite easy to drive nails just by using a wooden board pressed straight against your hand and hitting the nail - this spreads out the force of the blow over your hand, preventing pain and injury and allowing you to hit harder.
Finally, raw force is probably the dominant factor here. Pushing allows you to use the full strength of your muscle, which is probably somewhere around your weight (with a rather large spread). On the other hand, hitting allows you to accumulate the strength of your muscles over the duration of the swing, allowing you to impart much bigger forces than would be possible with just pushing. Try driving nails just by pushing the hammer, and you'll see the difference rather easily - the only benefit you'll get from using a hammer is that you're not going to feel as much pain as when pushing against the much smaller nail head.
A: This is in fact possible, as evidenced by the number of videos online of someone hammering a nail in with their bare hands.
More to the point, the human hand is plenty hard enough to hammer in a nail into some pretty hard if it is swung fast enough. It's just that the damage to the hand would be rather devastating, and so our pesky brains keep us from hitting a nail as hard as we can.
A: It is just because when you hammer a nail, you apply a force to the nail by changing the momentum of the hammer $F=\dot p$ This force helps in overcoming friction and thus pushing the nail inside. This isn't the case with your hand. Moreover, if you try to develop the same amount of force with your hand, you will get hurt because of the pressure applied to your hand by the head of the nail. Increased impact may also lead to the nail penetrating your hand, so be careful ;)
A: You can!  But first, some precautions:


*

*Your hand is squishier than the wood (generally, for wood of a condition to have new nails driven into it and hands prepared to do some hammering).  This means that it will take less pressure for the nail to damage your hand than for the nail to enter the wood.  The pointy bit at the front of the nail and the broad head at the end you are holding will help a lot.  However, it could be a good idea to spread the force around a bit more at the end you are going to be pressing on.  One way to do this is to glue a larger flat surface onto the head of the nail for you to press on.  I have these spare bricks...

*But all this gluing and drying takes too long.  Instead, let's just pick up a small block of waste wood, a 4" (10 cm) length of 2"-by-4" (5cm by 10cm) would be adequate.  Hold that mostly flat between your hand and the head of the nail and push.  See it works.  Only a couple thousand more to go until you're done building this house...

*Hmm...  Maybe this is still taking too long.  I know:  let's speed things up a bit.  Instead of just pushing against the nail, why not get a good run-up.  Raise the piece of waste wood up and lower it quickly onto the head of the nail.  (You'll probably have to hold the shaft of the nail in place with your other hand.  Warning:  you are about to swing a block of wood at your holding hand, so you might want to be ready to move it out of the way as soon as the block hits the head of the nail.  Some yelling may occur if you are not fast/careful enough.)  That's going much faster...

*But, unfortunately, even though the block of wood is spreading the force around on your hand, it's still receiving higher pressures during striking from the head of the nail.  This dents and scores the surface of the block.  I suppose we need to use something a bit tougher.  I happen to have this steel doorstop on me.  Try that.  Yup.  It's still spreading the pressure on your hand.  But rather than whacking with the flat part, it mugh be easire to get a good grip around the long axis and use it that way.  (This is because, the doorstop is so much thinner than the block that your fingers actually curl around to cover a little bit of the flat side we're hitting the nails with.  To prevent the occasional clipping and snipping of finger tips by imminently smacked nailheads, we might want to find a way to hold the doorstop that doesn't have our fingers on the striking face.)

*You know.  Holding that that way...  You could probably lift it above your head and swing it down onto the head of the nail and get a lot more momentum transferred to the nail.  Try it.  Yup.  That worked much better.

*Oops.  Now the wedge you're holding tries to pry your fingers apart on every hit.  It's also pretty good at transferring a jarring whack to your fingers on every hit which is making them numb.  Sure is inconvenient to have the same momentum dumped back into your fingers as you are putting into the nail.  What can we do?  I'll think on that.  Keep up with the framing and I'll go rummage around in my junk box and see if anything comes to mind.

*I had a great idea.  I've go this old, unused axe handle.  Notice that you're pretty much holding an axe.  This'll be great:  we attach the doorstop to the end of the handle and then you swing the contraption, kind of like an axe, to kit the heads of the nails with the small flat face of the doorstop.  This should really lower the pressure on your hand, since some of it will be appear as torque, and also the miniscule flex of the handle will spread the momentum transfer out a little bit in time and you won't be holding a wedge trying to drive your fingers apart.  This'll be great, I promise.


So, yeah, while you can drive a nail with your bare hand.  It's a much better idea to use a hammer.
A: Your hands are soft whereas the hammer head is very hard. Therefore your hand will come to rest upon striking a nail in a much greater time as compared to hammer.
Now, force is the rate of change of momentum. The hammer is heavier so momentum is more initially even if hand and hammer hit at same velocity. As well as the time taken for the momentum to change to zero is less in case of the hammer. Both these factors increases the force exerted by the hammer considerably. Hence, it is able to drive a nail inside a wall. 
A: TL;DR:
In order for the nail to be driven into the wood, you need to apply sufficient force to split apart the wood fibers ("make a hole") and overcome the force of friction between nail and wood.
The hammer can apply a much greater (instantaneous) force than the hand, because it is much harder (has a greater Young's modulus).

If you assume a mass $m$ is hitting a stationary object with velocity $v$, and there is a small spring between them with spring constant $k$, then we calculate the compression of the spring $\Delta x$ with a simple energy argument:
$$ E = \frac12 m v^2 = \frac12 k \Delta x^2$$
The maximum force $F = k\Delta x$ then follows:
$$F = \sqrt{m k} v$$
We see that the force goes up with velocity, and with the square root of either the mass $M$ or the spring constant $k$. In other words, a harder object (like a hammer) has a higher "spring constant" $k$, and because it decelerates over a shorter distance it will apply a greater force. And it's force we need to drive the nail.
Another way to think about this is in terms of conservation of momentum. If we need to lose momentum $Mv$, we can do this by applying a force $F$ for a time $\Delta t$. The shorter $\Delta t$, the greater $F$. Again, a "squishy hand" will spread the impact time further, and reduce the peak force.
There are secondary things at play here - for example, if you put your hammer at the end of a long stick (sledge hammer), you will be able to make it swing faster: and the force goes linearly with velocity, but only with the square root of the mass. Of course it's harder to hit accurately with a long-handled hammer: but it will deliver a bigger punch than a short, heavy one.
A: If you're strong enough, you can push the nail in by hand. You only need protection against the "equal and opposite reaction" of the nail against your soft hands. The nail can penetrate the wood because it is sharp, applying high pressure to the surface area it contacts. The nail's head is relatively much wider, so it will not penetrate your hand as easily as the sharp end, but it is clearly not engineered to be pushed in by hand. It's still too small a surface area for you to apply heavy pressure against your weak human skin.
To push the nail in by hand, you need some kind of defense against that pressure.


*

*You could wear an iron gauntlet, like a medieval knight. It will not be pierced and will distribute the pressure across a larger
surface area on your hand, preventing injury.

*You could place a large, at least hand-sized, metal disk atop the nail, essentially creating a nail with very large head. That's
    what a nail would look like if it were engineered for
    hand-force-touching. Of course, if you jumped on the disk you would
    probab
ly drive the nail in much faster than by just pushing on it.
A: Basics, Newton law and acceleration integration:
sum(F) = m.a
v = a.t

The force exerted by the hammer is dissipated in about no time, because the hammer is very though. The acceleration is therefore great.
Now, considering that your hammer is .2kg, going at 10m.s^-1, and hitting the nail during 1ms, it is pushing by (0.2kg*10m/s)/0.001s = 2000N
That's the amount of thrust required to lift 200kg.
In comparison, when you lift that much, assuming you are very strong, you rely on you whole hands. But with only one hand, most people will generate less than 500N, and without hurting themselves, with a singe finger, maybe less than a 100.
Now, the reason why pushing slowly on the nail does not work is the friction of the wood.
A: Because your hand is softer than a hammer, absorving most of the impact power over the nail, reducing overall power needed to penetrate wood. To compensate this, is necessary to apply too much more power that your hand will simply be injuried by nail. If the resistence of your hand skin is lower than resistence of wood, only your hand will be perfured, even considering the flat nail head format. However, if you got a nail with a ruge head, like it was a giant pin, you could tap with open hand and so got success. The bigest problem here, in truth, is the pression. The Hammer metal head can suport the nail little head size pression, your hand no. How smaller the surface area with same force, greater the relative pression. Then pression is the true force here, not exactly the hammer push strengh.
