How can the atmospheric pressure be different in distinct points at the same altitude? From an hydrostatic point of view, the pressure in a fluid should be the same at the same depth/altitude.
Obviously, in our atmosphere that does not happen. I am guessing that the main reason is the fact that the atmosphere cannot be regarded as hydrostatic.
Is this the reason? How exactly can we explain these pressure differences?
I understand that a higher pressure region must have a higher density, and therefore it would take time for reducing such density gradient. But how fast is this? In the order of the speed of sound? Or it has nothing to do with it?
 A: You asked a number of questions in this question.

From a hydrostatic point of view, the pressure in a fluid should be the same at the same depth/altitude.

That "should be" assumes hydrostatic equilibrium. That is a simplifying assumption. It's a reasonable starting point, but it's not a hard and fast rule. The Earth's atmosphere, it's oceans, and even its interior are approximately in hydrostatic equilibrium.

I am guessing that the main reason is the fact that the atmosphere cannot be regarded as hydrostatic. Is this the reason?

Significant deviations from hydrostatic equilibrium do occur. This is an effect, not a cause.

How exactly can we explain these pressure differences?

Ultimately, it's because the Earth


*

*Is round,

*Is lit by the Sun,

*Rotates about its axis once per day,

*Has distinct rotational and orbital axes, separated by about 23 degrees,

*Has a fairly clear atmosphere, and

*Is covered by lots of liquid water.


These result in climate and weather, which in turn result in the Earth's atmosphere being only approximately in hydrostatic equilibrium.
Equatorial regions receive a lot more sunlight than do polar regions. The resulting temperature gradient is one of the key drivers of the climate. On Venus, which rotates slowly, this energy transfer occurs in a pair of Hadley cells that reach from the equator almost to the poles. On Titan, which rotates in about 16 days, the Hadley cells breaks up at about 60 degrees latitude. Jupiter and Saturn are so large and rotate so quickly that they have bands instead of Hadley-type cells.
On the Earth, which rotates once per day, the Hadley cells extend to only 30 degrees. Polar cells form around the poles, and the Ferrel cells act as intermediaries between the Hadley and Polar cells. 
http://www.metoffice.gov.uk/media/image/f/s/Figure-4-Global-cells(edit)2.jpg

But how fast is this? In the order of the speed of sound? Or it has nothing to do with it?

The speed of sound has nothing to do with it. Winds do, and winds generally move much slower than the speed of sound. The fastest winds recorded are inside tornados, and even there things only move at about 40% of the speed of sound.
A: The air moves in great swirls.
In places where the air is being warmed from below it moves up.
That causes air to be sucked in from below, and spread out at the top.
What it sees as the reason to be sucked in is a lower pressure pulling it.
When any fluid is pulled in to a center, its angular momentum is conserved (and it has plenty of that because it is spinning with the earth), so it spins faster.
(Coriolis force is another way to describe this.)
So, you have meteorological low pressure areas, where the air is spinning the same direction as the earth, only faster, and high pressure areas, which are the opposite.
So that's why you can see different pressures at sea level or any other altitude.
(By the way, a low atmospheric pressure at sea level causes the water itself to be pulled up, resulting in "storm surge".)
A: A very simplified explanation: because the temperature is not everywhere the same. Why is the temperature not uniform? There are various reasons, the most important reason for temperature and pressure differences at locations not too far from each other is that the ground below is not the same everywhere. Depending on whether it's a forest, a lake, a field, or rocks below you, the ground absorbs and reflects heat differently. The humidity will also be different, depending on what is on the surface.
This local variation in temperature leads to air getting warmer and rising up at one location, and getting colder and moving downward at another location, leading to pressure differences. Air then moves around to equalize these pressure differences, this is what we call "wind".
A: Another example (in the question and not clarified by any response) of treating compressible and on-compressible fluids the same. Once they are separated the problem is simplified and for me at least sorts itself out.
