When I drink with a straw, is the atmospheric pressure pushing the water or me sucking it? Imagine a simple setup: me, a long cylindrical straw, a party cup filled with water.
When I drink from the straw, I like to know if the water inside the straw is being pushed or pulled? Which action is more dominant in this case? Is it the suction from my mouth to draw the water from the straw or the atmospheric pressure pushing on the water in the cup?
 A: Those are just different ways to name the same thing. Ultimately it is atmospheric pressure that pushes the liquid up the straw but normally the atmosphere wouldn't do that: the reason the water moves is because you created a low-pressure zone in your mouth which allowed atmospheric pressure to push the water up. If you define sucking as creating a low-pressure zone to move liquids or gasses then both options are correct: the water is being sucked up by you and it is pushed up by the atmosphere.
A: Atmospheric pressure is a result of elastic collisions between molecules, so the molecules bounce or "push" off one another. When you drink out of a straw, it's like a reverse tug of war where both sides push instead of pull. Before you start sucking, air inside and outside of the straw are at identical pressure - both sides "push" equally hard, and the liquid in the straw doesn't move. As you suck on the straw, you reduce the pressure within the straw, meaning it does not push as hard on the liquid. The forces on the liquid are now imbalanced since the air outside the straw pushes harder than air inside the straw, forcing the liquid up the straw.
Air pressure arises due to "pushes" between molecules. A vacuum does not "suck" - note there is literally nothing there to give rise to any sucking force. Rather, it is the fluid pressure that is always pushing, and if there is insufficient force pushing back, movement can occur due to the imbalanced forces.
Imagine if instead of sucking through the straw, you blow instead - the ambient air pressure around the glass doesn't suddenly change from a pushing force to a pulling force. Instead, the pushing force around the glass is simply overcome by an even larger pushing force inside the straw. But whether pressure in one area is higher or lower than another does not change the nature of the pressure force. Although pressure is fundamentally a pushing force, suction from your mouth is required to imbalance the pressure, which is why water does not get pushed up a straw spontaneously.
A: It's atmospheric pressure. This has an experimental consequence: suction can't lift water more than about 10 meters. This is because the pressure at the bottom of a 10 meter water column with vacuum above it matches atmospheric pressure, so the atmosphere can't push it any higher.
This was known in Galileo's time, and he attempted to explain it, but didn't get it right.
A: This question is like asking "When a ball falls, is it moving away from the higher level or moving towards the lower level? Does it move because the higher level is higher, or because the lower level is lower?" It's the same, both levels (atmospheric and lung pressure) are equally important. It is their difference that drives the flow (of ball or air), not solely the value at one point or the other. If they have different values, there will be a gradient, and things tend to flow down gradients. This is a universal phenomenon with many examples: pressure gradient (fluid movement), gravitational energy gradient (objects falling), electric potential gradient ("voltage", electricity flowing), concentration gradient (diffusion/particles moving in a fluid) and so on. So the sucking and pushing are two sides of the same coin, and the coin (fluid flow) would not exist without two sides. It is best to think of it not from one side or the other (lung sucking or atmosphere pushing), but as a net movement created by a pressure gradient/difference, which is a property of how the pair of points relate to each other.
A: What is traditionally thought of as sucking simply doesn't exist, at least in the way most people naively think about it.  Removing the straw, the situation is identical to what happens when you take a big breath of air and your chest/lungs puff up. Most people would probably say this is because you are sucking in air and that air is causing your lungs and chest to expand.  This is incorrect.  The causal relationship is reversed, which is why I think viewing pushing and pulling as the same thing as some answers suggest isn't really correct -- there is a clear time ordering to what is occurring.  First the muscles in your body causes your lungs expand which creates a pressure differential and this expansion* makes it so the air around your nose/mouth rush into your body so that your lungs have the same pressure as the outside air.  Nothing is being pulled, molecules of air are being pushed into you by other air molecules.  The only difference between this and drinking through the straw is that the fluid under discussion is water and not air, other than that nothing conceptually changes.
*a simple application of the ideal gas law, PV=nRT .  The RHS of the equation doesn't change so an increase in volume corresponds to a decrease in pressure inside the lungs.
A: It is the atmospheric pressure pushing it. In a vacuum, if you tried sucking on a straw nothing would happen (you wouldn't even be able to suck). Or if you sealed a rigid container so the only opening was the straw nothing would happen either. Rigid so the atmospheric pressure cannot deform the container to interact with the liquid inside.
Your question is like asking, "If I scoop water from a pool and more water around it flows in to fill its place, is it me scooping water that causes the water to flow? Or the weight of the water remaining in the pool?"
In both cases, your action caused an imbalance in the pressure, but the flow itself is caused by the pressure. If the source of pressure is not there whether it be gravity or atmospheric pressure, your actions would do nothing.
A: It is the atmosphere pushing the surface of the liquid. The energy transferred to the liquid is equal to the volume displaced times the ambient air pressure.
You do not get this energy for free though. Every joule you get from the atmosphere you have to give back to the atmosphere by pushing some part of your body against it. If you're sucking with your mouth sealed off of your lungs, it is the underside of your chin pushing against the atmosphere. If you're sucking with your lungs, it is your chest and abdomen. Either way, it's the atmosphere in contact with your body that you're fighting when you suck. That's where the energy goes.
A: Yes and no.
To drink from/with a straw, you first need to make an airtight seal with your mouth around the straw.
You then use the muscles in your diaphragm and possibly rib cage to expand your chest cavity.  This lowers the pressure inside your lungs, so carrying out this expansion requires that you to do work against atmospheric pressure. This energy is stored in the reduced lung pressure
Then the pressure on the top of the liquid in the straw is lower than the pressure of the atmosphere on the free surface of the liquid in the bottle/ glass.  So the excess atmospheric pressure does work on the liquid, forcing the liquid to rise in the straw.  If the straw is short enough, or your diaphragm is powerful enough, the liquid level reaches your mouth and Ta-da!
Incidentally, you also need to seal off your nostrils (internally!) or you will suck in air through your nose, not liquid through your straw
A: Sucking on the straw reduces the pressure at the top of the straw allowing atmospheric pressure (plus the pressure from the weight of the liquid at the bottom of the straw) to create sufficient force to push the liquid in the straw up into your mouth.
A: Hydrostatic model 
For simplicity we consider a vertical straw with a tip at depth $h$ beliw the surface of the liquid and the level of liquid inside the straw reaching height $H$ (measured from the submerged tip). This column of liquid has weight $\rho g H A $, where $A$ is the cross sectional area of the straw; the force acting on it from the bottom is proportional to the pressure at depth $h$: $(\rho g h + P_{atm})A$, whereas the force pushing on it from the top is die to the pressure created by inspiration of the air in the straw: $AP_{insp}$. The force balance is thus written as
$$(\rho g h + P_{atm})A=\rho g H A+AP_{insp} \Leftrightarrow 
\rho g h + P_{atm}=\rho g H +P_{insp}.
$$
The maximum height to which one can draw a liquid by inspiration (i.e., inhaling/suckung air) is
$$
H=h + \frac{P_{atm}-P_{insp}}{\rho}.$$
Estimates 
We can now do the estimates: 

*

*The atmospheric pressure is 1atm, corresponding to 1033 cm column of water


*Maximum inspiratory pressure (MIP) - that is the maximum pressure created by inhaling - is about 130 cm of water for men and about 100 cm of water for women. 
In our case the MIP actually corresponds to the difference $P_{atm}-P_{insp}$ - that is one can suck water or a similar liquid via a straw about one meter long.
Further remarks 

*

*One can suck repeatedly, effectively creating vacuum in one's mouth. In this case the maximum length of the straw is limited by the atmospheric pressure, i.e. about 10 meters. Some internet pages report actually checking this experimentally, as a part of physics class.


*In practice the situation is rarely static, so one might try to make a correction using Bernoulli  law, which means that, once the liquid is moving up the straw, the pressure needed to keep it moving is smaller than in the static case.


*Capillary /surface-tension effects - one can observe these by looking down into a straw. The liquid rises a millimeter or so, so the associated force is just too small for a typical straw diametwr to be of any importance.
