Technically, this word refers to difficulty or inability to stop an activity. This activity can be an idea, problem solving strategy, word, movement, or complex activity. People who perseverate may get stuck on a narrow train of thought, or in the beginning of a movement pattern they can't complete. They may repeat words, phrases, or parts of words. They may be unable to change problem-solving strategies, even when the one they are using is clearly not working. This last form is measured by the Wisconsin Card Sorting Test.
Neurologists associate perseveration with damage or abnormal functioning in the the frontal lobes. It can be seen to varying extents and durations in a variety of neurological conditions, including but not limited to frontal lobe brain damage, frontal lobe seizures, autism and other pervasive developmental disorders, dementia, attention deficit disorder, delirium, catatonia, obsessive-compulsive disorder, Tourette's syndrome, and various legal and illegal drugs. If someone with suspected mental illness is perseverating, it may point to an organic cause of their unusual behavior.
One should be aware that when autistic people say they're perseverating, it doesn't necessarily mean they don't enjoy it or that they want to stop. Autistic people's narrowly focused interests have been labeled perseverative, so some people have taken on a more idiosyncratic use of the term. They use perseveration to refer to an often-enjoyable interest or activity that one can focus on for extended periods to the exclusion of other activities. At the point where such an activity becomes unpleasant and unstoppable, they call it obsession, compulsion, or inertia. Autistic people do, however, experience both the formal and informal kinds of perseveration, and they are in all likelihood related to each other.
Depending on the severity of perseveration, it may be treatable by behavioral and cognitive strategies, or sometimes by medication.