What exactly is pressure? Say, I have a balloon full of gas and the gas is exerting pressure on the balloon wall. So, what exactly are the gas atoms/molecules doing to the balloon wall?
 A: If you treat the gas as an ideal gas (which is a fairly good approximation in most situations), you can model the gas as a collection of molecules which are moving randomly and are experiencing ellastic collisions between each other. Then, if you put the gas into a container, the molecules also collide randomly with the walls of the container. As there is a lot of (I mean a very lot) molecules in any macroscopic-size container, you can approximate the effect of these collisions as if the gas exerted some force on the walls of the container.
Now, pressure is, according to the definition, just the ratio of force to the area of surface it is exerted on:
$$
p = \frac{F}{A}
$$
Thus, the pressure of a gas put into a container would just be the force exerted by the molecules of the gas onto the walls of the container, averaged over the total area of the walls.
A: For simplicity if we take a single collision between a molecule of gas moving perpendicular to a wall of the vessel we see there is a change in momentum of $2mv$ where $m$ is the molecule's mass and v is its speed, assumed constant, and this is equal to the impulse provided by the wall equal to  $F\delta t$ where $F$ is the force on the molecule due to the wall and $\delta t$ is the time over which it acts.
The problem is for a single molecule we don't know $\delta t$. This is where the kinetic theory breaks down unless you are able to make the assumption that we can consider the force due to the wall acting continuously over a longer time period $T$ and over that time period we look at the total change in momentum of the particle due to the wall. If we know that the the distance between opposite walls of the vessel is $L$ then clearly in time $T$ we have approximately $Tv/2L$ collisions and hence a total momentum change of $$2mvTv/2L=Tmv^2/L=FT.$$ Cancelling the $T$ we get $F=mv^2/L$. And this is the starting point for the full derivation of the gas laws since pressure is defined as force per unit area $=F/L^2$ and taking the vessel to be a cube with sides $L$ we have pressure on the wall due to the molecule (by Newton's second law since the wall is not moving given by $$P=mv^2/L^3=mv^2/V=kT/V$$ for some constant $k$, where $T$ is the temperature of the gas defined to be proportional to the average kinetic energy of the gas molecules.
