Altitude sickness is by far the dominant health issue on the mountain and really the only thing that makes climbing Kilimanjaro so difficult. Kilimanjaro is seriously high. No amount of training and no amount of other trekking below 4000m altitude can prepare you fully for this. Almost all people who climb Kilimanjaro suffer some form of mild to medium symptoms of altitude sickness.
Some trekkers suffer symptoms severe enough to warrant their immediate removal to lower altitudes. Up to 1% of trekkers require emergency evacuation. Altitude sickness does need to be respected. Following the advice below should minimise the chances of inconvenience and catastrophe to acceptable levels.
Unlike other extreme peaks such as those in the Himalayas, the summit of Kilimanjaro is approached in just a few days, leaving very little time for the body to adjust. This is what makes Kilimanjaro such a unique, difficult and potentially dangerous challenge. A general rule on really big mountains like this is that you should climb high and sleep at low altitudes whilst ascending up Kilimanjaro.
High altitude: 2400m to 4200m This zone covers the majority of all treks on Kilimanjaro, except the first half day on some of the lower starting routes. Here you can expect to be seriously impacted by the effects of altitude in terms of workload and stamina and you may start to feel the symptoms of sickness such as headaches and nausea.
Very high altitude: 4200m to 5400m This zone covers the night before summit attempt on all routes and, on better routes, one or two acclimatisation hikes on earlier days. Here the effects of altitude are starting to become more acute, walking at anything other than a snail's pace is quickly tiring. Most people will now be experiencing adverse symptoms unless they are taking supplementary oxygen. Some people cannot handle this altitude at all and will need to be removed from the mountain.
Extreme altitude: above 5400m This zone covers much of the summit attempt, with Uhuru Peak at 5895m. At these altitudes your body is simply deteriorating, if you stay up here too long without oxygen you will die. Unless you have supplementary oxygen you are basically making a run for the summit in order to get down before you succumb to the adverse effects. If you have oxygen then you can take your time and will be more likely to clearly appreciate what is going on.