Why doesn't the moon twinkle? Stars twinkle because their light has to squeeze through several different layers of the Earth's atmosphere. So why doesn't the moon twinkle as well?
The first handful of hits on Google actually return incomplete and even wrong answers (e.g. "Because the Moon is much brighter" which is plain wrong, and "Because the Moon is closer" which is incomplete [see below]). So here's the answer:

As you mention, when light enters our atmosphere, it goes through several parcels of gas with varying density, temperature, pressure, and humidity. These differences make the refractive index of the parcels different, and since they move around (the scientific term for air moving around is "wind"), the light rays take slightly different paths through the atmosphere.

Stars are point sources

Stars are immensely far away, effectively making them point sources. When you look at a point source through the atmosphere, the different paths taken from one moment to another makes it "jump around" — i.e. it twinkles (or scintillates).

The region in which the point source jumps around spans an angle of the order of an arcsecond. If you take a picture of a star, then during the exposure time, the star has jumped around everywhere inside this region, and thus it's no longer a point, but a "disk".

…the Moon is not

The same is true for the Moon, but since the Moon (as seen from Earth) is much larger (roughly 2000 times larger, to be specific) than this "seeing disk" as it's called, you simply don't notice it. However, if you are observing details on the Moon through a telescope, then the seeing puts a limit on how fine details you can see. 

The same is even true for planets. The planets you can see with the naked eye span from several arcsec up to almost an arcmin. Although they look like point sources (because the resolution of the human eye is roughly 1 arcmin), they aren't, and you will notice that they don't twinkle (unless they're near the horizon where their light goes through a thicker layer of atmosphere).

The image below may help understanding why you see the twinkling of a star, but not of the Moon (greatly exaggerated):





EDIT: Due to the comments below, I added the following paragraph:

Neither absolute size, nor distance is important in itself. Only the ratio is.

As described above, what makes a light source twinkle depends on its apparent size compared to the seeing $s$, i.e. its angular diameter $\delta$ defined by the ratio between its absolute diameter $d$ and its distance $D$ from Earth:
$$
\delta = 2 \arctan \left( \frac{d}{2D} \right)
\simeq \frac{d}{D}\,\,\,\mathrm{for\,small\,angles}
$$

If $\delta \lesssim s$, the object twinkles. If it's larger, it doesn't.

Hence, saying that the Moon doesn't twinkle because it's close is an incomplete answer, since for instance a powerful laser 400 km from Earth — i.e. 1000 times closer than the Moon — would still twinkle because it's small. Or vice versa, the Moon would twinkle even at the distance it is, if it were just 2000 times smaller.



Finally, to achieve good images with a telescope you not only want to put it at a remote site (to avoid light pollution), but also — to minimize the seeing — at high altitudes (to have less air) and at particularly dry regions (to have less humidity). Alternatively you can just put it in space.