Anxiety is a sense of heightened awareness in response to stimuli. In addition, respiration and heart rate often increase, usually in response to the fight or flight response typical of an increase in adrenaline. All people feel anxiety, but many individuals have heightened anxiety, in response to specific stimuli that generally do not raise anxiety in humans. For example, it's normal to be anxious around your boss, but it is not typical to fear high levels of anxiety around cats, although some people do. A crippling sense of anxiety or fear in response to a specific stimuli, particularly those commonly found in social situations, is known as a phobia.
Anxiety about one's health can lead to psychosomatic illness, as well as a panic attack.
Although occasional anxiety is normal and extreme anxiety usually has no long term effects, long term anxiety usually has several adverse health effects, generally due to the effect adrenaline has inside of the body. This includes insomnia, lack of appetite, and high blood pressure.
High levels of anxiety can be treated as an illness with anti-anxiety medication and psychotherapy, but anxiety is not in and of itself an illness, even if phobias are.