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?

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.

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 ;)

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.