Humans can survive up to about 20,000 ft (3.8 miles) above sea level. Above about 20,000 ft, the human body begins to suffer a condition called hypoxia in which the brain does not receive sufficient oxygen for long-term survival. The problem is that above 20,000 ft the air density is too low.
The air at 20,000 ft still contains about 21% oxygen, but the number of air molecules per volume is too small. To understand this, you need consider how breathing works. When you take a breath, muscles work to expand your lungs (increase the volume in your lungs). Air rushes into your lungs until the air pressure inside your lungs equals the air pressure outside (the atmospheric pressure). Another way to think about it is that air moves into your lungs until the air density (number of molecules per volume) inside is about equal to the air density outside. (For all you science majors, the air density inside your lungs will not equal the air density outside because the air in your lungs is at a different temperature than the air outside, but it helps with the explanation).
You cannot compress air in your lungs to a higher pressure (or density) than the air outside. Your body extracts oxygen from the air whenever an oxygen molecule collides with the oxygen-accepting surfaces of your lungs. At high altitudes, the air density decreases, so there are fewer air molecules in the volume of your open lungs. There are just not enough collisons between oxygen molecules and the surface of your lungs to extract sufficient oxygen for bodily functions. Many people, especially those not accustomed to high elevation, begin to experience problems due to oxygen deficiency at altitudes well below 20,000 ft (altitude sickness).
What about commercial jets which often fly at altitudes above 30,000 ft? Commercial airplanes are pressurized, i.e., outside air is brought in and compressed in the cabin, raising air density and pressure. The announcement "the cabin has been pressurized to a comfortable altitude" means that the cabin pressure has been adjusted to be the same as the atmospheric pressure at a lower altitude (than the plane is flying). Any hole in the cabin results in air rushing out of the plane until the cabin pressure equals the outside pressure. If this occurs, oxygen masks must be used or you will not be able to live.
Rapid changes in outside air pressure can cause the discomforting feeling of ear popping. Most often the pressure changes occur when ascending or descending through the atmosphere. The physiological details about ear popping are described in this Ear-Popping web page.