How come gas molecules don't settle down?
The key ingredient is temperature.
If it were zero then all the air would indeed just fall down to the ground (actually, this is a simplification I'll address later). As you increase the temperature the atoms of the ground will start to wiggle more and they'll start to kick the air molecules giving them non-zero average height. So the atmosphere would move a little off the ground. The bigger the temperature is the higher the atmosphere will reach.
Note: there are number of assumptions above that simplify the picture. They are not that important but I want to provide a complete picture:
- Even at the zero temperature the molecules would wiggle a little because of quantum mechanics
- the atmosphere would freeze at some point (like 50K) so under that temperature it would just lie on the ground
- I assumed that the ground and the atmosphere have the same temperature because they are in the thermal equilibrium; in reality their temperatures can differ a little because of additional slow heat-transfer processes
To leave you with something a little bit more concrete (although quite simplified), statistical mechanics tells us that probability $p$ that some portion of the system will occupy a given energy $E$ level is $p \propto \exp({-E \over kT})$ where $k$ is a Boltzmann constant (note that you have to normalize this probability to 1 by summing over all energy levels; that's why it's only proportional, $\propto$).
For the atmosphere of ideal gas where molecules have only potential energy (kinetic energy is not important here because the molecules don't interact with each other in the ideal gas) we have $E = mgh$. This gives us $p \propto \exp({-mgh \over kT})$. You can see that atmosphere "wants" to get lower ($p$ is bigger for $h$ lower) but the temperature "forces" it to go up. Play with the equation for different values of $T$ and $g$ (gravity strength) to see what happens.
First of all, gravity does continually accelerate the air molecules. I don't see how this could make them lose their momentum.
What is the net effect of gravity on the atmosphere? Simple, gravity prevents the atmosphere from flying off in space, and instead it keeps it comfortably wrapped around our planet!
The reason why the atmosphere is still thick after billions of years is because you have two net effects on the air molecules, gravity, which keeps it as close as possible to the ground, and inertia, who has the opposite net effect. So as long as the molecules do not slow down they "orbit" our planet.
The reason is the same as why is the moon orbiting the Earth after billions of years. There's a balance between the kinetic energy of the moon and the gravitational potential energy - or a balance between gravity and inertia.
The other answers give you a summary explanation of what determines the temperature of the atmosphere and hence its molecules' average velocity. The reality is way more complicated as the temperature of the atmosphere is not constant with height and you have to take into account many more factors like varying pressure, convection and so on. Modelling the Earth's atmosphere accurately is very complicated.
In conclusion the basic mechanisms are outlined above. I hope they answer your question.
The other answers are correct but to understand them you have to get an idea of how much thermal energy does an average molecule have.
According to Maxwell-Boltzmann distribution, the most probable speed of an air (say, nitrogen) molecule at room temperature is $v_p = \sqrt { \frac{2kT}{m} } = 422 m/s$. Without collisions with other molecules it can travel upwards $h=\frac{v_{0}^{2}}{2g}$ = 9 kilometers before the gravity stops it and pulls back to Earth. Basically, potential energy of molecules in gravitational field is too small compared to their kinetic energy to keep them low.
Update: Still, gravity is the reason why we have atmosphere after billions of years.