In Kenya, a dispensary is a small outpatient health facility, usually managed by a registered nurse. It provides the most basic primary healthcare services to rural communities, e.g. childhood immunization, family planning, wound dressing and management of common ailments like colds, diarrhea and simple malaria. The nurses report to the nursing officer at the health center, where they refer patients with complicated diseases to be managed by clinical officers.
I assume it is much the same in Tanzania, although the bureaucratic structure could well be different.