Bipolar disorder is when someone has symptoms of major depression plus symptoms of mania which is a period of abnormal, and persistent elevated, expansive and irritable mood. The person alternates between depression and mania. So, yes it is an illness. There are very specific criteria that must be met to be able to diagnose a person as having bipolar disorder. Also, this is associated with changes in the levels of some chemicals in the brain and so drugs are given to normalize the levels of these chemicals.
From what I know people with psychiatric problems don't really know that they have one. They are not aware that they have a problem. Only people who are close to them notice the changes in behavior.