Insulin sensitivity is a general phenomena in the body, and can be measured a few ways through studies.
The pancreas (an organ that regulates blood sugar) secretes insulin in response to high blood sugar, and cells (like muscle or fat cells) can absorb blood sugar when stimulated by insulin.
Insulin sensitivity is the relationship between how much insulin needs to be produced in order to deposit a certain amount of glucose. You are insulin sensitive if a small amount of insulin needs to be secreted to deposit a certain amount of glucose, and insulin resistant if a lot of insulin needs to be secreted to deposit the same amount of glucose.
Insulin sensitivity is seen as good as the opposite, insulin resistance, is a major risk factor for the development of Type II diabetes.
There are three main types of insulin sensitivity; peripheral insulin sensitivity, hepatic insulin sensitivity, and pancreatic insulin sensitivity.
Peripheral insulin sensitivity is how readily body cells in your periphery tissue, such as muscle and fat, can absorb glucose; either on their own (muscle can absorb glucose when contracted) or when insulin stimulates them. It is the most well-known form of insulin resistance.
Hepatic insulin sensitivity is related to the process of gluconeogenesis, the production of new blood sugar. Usually inflammatory factors prevent insulin from acting in the liver via inducing insulin resistance, and insulin's actions are unable to tell the liver to 'stop' producing glucose.  
Pancreatic insulin sensitivity is the functioning of the cells that secrete insulin, the beta-cells. If these are damaged or cannot function, insulin resistance can develop. This is more of a concern in disease states like Type I diabetes (insulin insufficiency)   or Cystic Fibrosis (where the function in physically hindered). 
Insulin sensitivity is how effective the body is as using insulin to reduce elevated blood glucose levels, with a greater efficacy being more 'sensitivity' and poorer efficacy being more 'resistant'. When the body becomes too poor at using insulin to reduce blood glucose levels, type II diabetes ensues
It seems that insulin sensitivity is negatively associated with age  although these may be related more to lifestyle than to age per se.  The ability to reverse insulin resistance with exercise does not appear to be different between young and old.  Exercise tends to be recommended to older individuals to improve glucose metabolism. 
There is an association with obesity and insulin resistance, with insulin resistant individuals usually having more body fat. However, this also appears to be lifestyle related as increases in insulin sensitivity can occur without weight loss.  Some studies do note more drastic benefits in insulin sensitization in obese individuals, which is probably due to worse baseline statistics. 
Aerobic exercise, or exercise that you can maintain for a prolonged period of time, seems to be able to acutely improve insulin resistance by increasing uptake of glucose into cells. It can increase insulin sensitivity immediately, as a session of 25-60 minutes (at 60-95% VO2 max) for 3-5 days.  Improvements can also be seen after a week of aerobic training, when doing mostly 2 short sessions of 25 minutes of walking at 70% VO2 max.  Interestingly, the opposite is also true. Voluntary restriction of activity or a drastic increase in sedentary activity can reduce insulin sensitivity in as little as 2 weeks. 
Over the long term, aerboic exercise done routinely can preseve beneficial changes in insulin sensitivity. 
Insulin sensitivity as a result of exercise can occur independent of weight loss.   This is not to say that aerobic exercise will not lead to weight loss, as it may.  The function of weight loss seems to be a blend of activity and diet, whereas insulin sensitivity increases could occur without changes in the diet.
Strength exercises (lifting weights usually) is also associated with increasing insulin sensitivity as well as increased muscle mass. 
In persons with impaired glucose tolerance, more sets of an exercise tend to be more effective than single sets and higher intensities better than moderate. 
The general idea of exercise is that you want to have lean (muscle) mass, and you want it to contract somewhat regularly so it can take up glucose. The more properly functioning muscle mass one has, the better peripheral insulin sensitivity is
We have a constantly expanding meta-page for insulin sensitivity that collects promising supplements that may increase insulin sensitivity.
These supplements may be either supplements that directly act upon cells to induce insulin sensitizing effects (like resveratrol or carnitine) or may inhibit or otherwise delay carbohydrate uptake (like green tea catechins and perhaps chlorogenic acid)
Using some of these compounds in conjunction with diet/exercise techniques conducive to regaining insulin sensitivity would be advisable.
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