Not sure if you asking about extreme elevations or at higher elevations in general, or if in relation to acute reactions or symptoms or how your body adapts to the lack of oxygen, but will offer some of my general observations, since I live at about 10,000 feet.
Generally after living at altitude for a couple of weeks, your body begins to produce more red blood cells to carry more oxygen, to compensate for the lower oxygen content of the air. Even with the higher red blood cell count, I find my aerobic output is less than it would be at a lower elevation, meaning I become 'winded' at a much lower physical exertion level than I would at a lower elevation.
Some of the things I see in people who come here and have some kind of reaction to the altitude is headache (very common), shortness of breath (again very common), and dizziness ( we have had people pass out here from altitude, but usually there are other factors involved in combination with the altitude). To a lesser degree, I see people who become queasy to the point of throwing up. Being dehydrated seems to aggravate the problem, and with the dry air here, dehydration sets in much sooner than people realize.
Some of the things that seem to help are aspirin (not Tylenol or Advil) because of the blood thinning attributes, drinking lots of water, and taking the time to acclimatize. Most people start out doing to much to fast, and it takes a toll pretty quickly.
Dave