Hemiplegia

Hemiplegia (or hemiphlegia) is a condition where a vertical half of a patient's body is weak or paralyzed, i.e. one arm and its corresponding leg do not function properly. It can be congenital (occurring before, during, or soon after birth) or acquired (as from illness or stroke). It is usually the result of a stroke, although disease processes affecting the spinal cord and other diseases affecting the hemispheres are equally capable of producing this clinical state.

Cerebral palsy can also affect one hemisphere, resulting in limited function. This does not cause paralysis but instead causes spasms. Cerebral palsy where this is the only symptom is often referred just as hemiplegia.

Hemiplegia is similar to hemiparesis, but hemiparesis is considered less severe. 

Reference

 * Patten C, Lexell J, Brown HE. Weakness and strength training in persons with poststroke hemiplegia: Rationale, method, and efficacy. J Rehab Res Dev 2004;41:293-312. Fulltext. PMID 15543447.