Is it catatonia, dementia, or retardation if someone is slow or sluggish?

Unclear: One would need lots more information to answer that. Mental retardation or developmental delays are life long and start in childhood, catatonia is associated with mood disorders like major depression or other neurological illnesses while dementia syndromes involve loss of functioning, memory impairment and other impairments usually strarting in old age.

Being slowed: Or sluggish is just a symptom, not a definition of an illness. Many medical problems can cause fatigue with one of the biggest offenders being hypothyroidism. First see you family doctor and if medically cleared, make an appointment with a psychiatrist in your area.