Do air particles "fly"? If not, how do they stay afloat? I was reading my old physics textbook (from middle school), and it mentioned something about the idea of having non-existing attractive forces between particles like air. "We would live in a very dull world."
This made me wonder, what would've happened if there are no bonds between air particles, or what if air particles stop moving entirely one day?
Will all the air particles just sink to the ground? (pulled by gravity)
Hence, the question: how do air particles "stay afloat" in the first place?
 A: @Steeven's point about there not being enough space on the ground can also be described in terms of there not being enough energy states near the ground. To take another familiar example, when you fill a bathtub the water "stacks"; some molecules go on the bottom, so go above them and have slightly more GPE, and so on. (If you flesh out this idea further, with energy to be general rather than altitude, you get to Fermi levels for particles in a suitable electrostatic potential.)
The occupation probability at energy $E$ is proportional to $\exp(-\beta E)$, with $\beta$ the thermodynamic beta. The approximation $E\approx mgh$ for mass-$m$ particles makes this factor $\exp(-\beta mgh)$, so air thins exponentially with length scale $\frac{1}{\beta mg}=\frac{k_BT}{mg}$, which works out to a few kilometres. This is why air pressure is lower at the top of a mountain. Unsurprisingly, this makes carbon dioxide thin faster (i.e. with a shorter length scale) than argon, which thins faster than oxygen, which thins faster than nitrogen.
A: I will list your questions and answer them one by one.

*

*what if air particles stop moving entirely one day?

This scenario is what happens when the temperature is very low. For really no motion at all you would need absolute zero temperature. But well before you get to absolute zero you get to another case: the gas turns to liquid, and then, when colder still to solid (except for special cases such as helium). Forming a liquid usually involves the attractive forces between molecules, but even if there were no attractive forces, the gas would eventually form a type of liquid. It would then lie in a big pool on the ground (while we all die for lack of oxygen).


*Will all the air particles just sink to the ground? (pulled by gravity)

yes, see previous ans.


*Hence, the question: how do air particles "stay afloat" in the first place?

They stay afloat through collisions. All the particles are indeed falling down owing to gravity, but they also bump into one another. You might guess that after a while they would on average sink lower and lower, but what happens instead is that there are more particles, that is, a higher density, at the bottom than at the top. And the ones at the very bottom do not sink any lower because they bounce off the ground. If they stuck to the ground then the whole atmosphere would itself fall and fall until it was all stuck to the ground. But they bounce off, and thus they provide a layer of gas near the ground. This layer then supports the one above it, because of collisions: the particles arriving from above get bounced back up again. And that layer in turn supports the one above it. And so on.
So the whole atmosphere is dynamic: between collisions every particle has a downward acceleration. During collisions the two particles bounce off one another. There is a higher density lower down, which results in more upward-directed collisions for a downward-moving particle than an upward-moving one.
All this can be captured precisely in equations, but I guessed you preferred the picture in words.
3B. But what if the molecules in the air did not collide with one another, only with the ground. Would the atmosphere fall down then?
This is an added paragraph suggested to me by some helpful comments by nanoman. He points out that in the scenario where the molecules do not collide with one another, they would still fly up high into the atmosphere after bouncing off the ground, following huge parabolas around 10 kilometres high, and overall the density distribution would still be the same! In this case the atmosphere thins as you go up because there are fewer molecules with enough energy to get that high. The above discussion in terms of layers is appropriate for the actual atmosphere because on average the molecules only travel tiny distances (less than a micron) before colliding.
P.S. I would like to add that the word 'bounce' is not quite right for what happens when air molecules hit the ground. In fact they mostly arrive and stick for a very short time called the 'dwell time', and then they get kicked or shaken off and zoom off in a random direction. The energy of the molecules coming away from this process is on average equal to the thermal equilibrium energy with which they arrived. So after averaging over time the net effect is like bouncing.
A: In short, the air molecules stay afloat because they are bouncing off the ground and other air molecules. Here's a video visualizing that with a simple simulation based entirely on kinematic principles: https://www.youtube.com/watch?v=vwk4mSFFop0
A: What seems to have remained unmentioned in the other answers is that the air moclecules do fall to the ground: the air at higher altitudes is both more diluted and colder, which is the reflection of the trade-off between the kinetic and the potential energy of the molecules (understanding that the average kinetic energy of the molecules is temperature, and that only high energy molecules manage to climb very high).
Update
To expand a bit in view of the discussion that followed:
Barometric formula predicts that the atmospheric pressure decreases with the altitide. The formula is derived assuming that the atmosphere is in equilibrium, i.e., it can be characterized by Boltzmann temperature and constant distribution (isothermal atmosphere), so that the average energy of each molecule is $$\langle E\rangle=\langle\frac{mv^2}{2}\rangle+\langle mgh\rangle = \frac{3}{2} k_B T$$
(neglecting the rotational and vibrational degrees of freedom) and hence the kinetic energy decreases with height - the molecules "fall" to the ground.
In reality the atmosphere is not at constant temperature and not in thermal equilibrium: the bottom of the atmosphere is at higher temperature than its upper layers, and the warm air constantly rises, while the cold air "falls". Within hydrostatic approach this is modeled as adiabatic atmosphere, resulting in the equation for the temperature variation with the altitide, see Lapse rate. There is a good discussion of the adiabatic atmosphere in this thread.
A: They stay apart because they are moving. A typical speed for an air molecule in the atmosphere would be 450 metres per second (rather faster than the speed of sound). When they hit each other they bounce off each other.
How do we know that the molecules are moving? Suppose that we pump some air into a closed container. We can detect that the trapped air exerts a pressure on the inside of the container. This is just what would happen if fast-moving molecules were continually hitting it. But, you might say, there  could be other causes of the pressure. Better evidence of molecules' motion is needed. Such evidence would be Brownian motion, in this case the observed jiggling motion of particles (like pollen grains) large enough to see under a microscope, in air. [The molecules of the air are far too small to see under a microscope, but they jostle the larger particles that we can see.]
What makes the air molecules move like this? They do it naturally. Scientists have known for a long time that temperature is a measure of how fast gas molecules are moving, or to be more accurate, of the mean kinetic energy of the molecules. And this is sustained by energy radiated from the Sun. If the temperature dropped very, very low the molecules would almost stop moving and pile up on the ground. [Of course liquefaction would take place, but that's another story.]
