Nothing really happens ... if you have a container and remove all the air from it, the atmosphere on the outside will press in on the container. If the walls of the container are strong enough, then the walls can support the imbalance of forces and nothing happens. On the other hand if the container walls are not strong enough the can will be crushed if sealed.
The air around us is pressured (called atmospheric pressure) and you can see its force by a simple experiment in which a small amount of water is boiled inside a can of coke and then the can turned upside down into a tank of water. As the water vapor that has displaced the air in the can is condensed (turned into water), the pressure inside the can drops. But the pressure outside the can is still atmospheric pressure. This excess external force crushes the can. Why doesn't air pressure crush us? The pressure inside our body is roughly the same as the pressure outside and therefore there is no net pressure (and thus, force) on us.
Taking the air out of a space is also called creating a vacuum. That is, if you don't replace the air with something else. What happens when you try to take the air out of a space depends on the space. The reason for that is that air from outside the space wants to rush in to fill the vacuum, to have the same amount of air everywhere. So if you take the air out of a soft material, such as a plastic bag, it shrivels up because the air all around it is trying to get in and deforms the bag. If the space is enclosed in a solid material, the container will not be deformed and you can really create a vacuum. Examples are TV screens that are not flat screen TVs. Those are made from glass and have a vacuum inside, which is why they can violently break or "implode" when the glass is damaged. Another example is a vacuum cleaner, where a vacuum is created by a fan, and the air that is rushing in to fill is pulls dirt up with it. The dirt then gets caught in the filter bag of the vacuum cleaner.
On a side note, it is pretty much impossible to take ALL the air out of a space. That would be a perfect vacuum. Depending on how much of the air one can remove, one may create a more or less high vacuum. Different levels of vacuum are required for some science experiments in which air or any gas would create problems. High vacuums require very strong pumps and that the space is extremely well sealed, which gets more difficult as the vacuum is stronger because it pulls air in more strongly.
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