A classic example is the anopheles mosquito which acts as a vector for the disease malaria by transmitting the malarial parasite plasmodium to humans. In this case plasmodium is harmless to the mosquito (its intermediate host) but causes the disease malaria in humans (its definitive host).
In molecular biology and genetic engineering a vector is a vehicle for transferring genetic material into a cell.
A viral vector is a virus which has been modified to transduct specific genetic material into a cell, e.g. for gene therapy.
A plasmid vector is made by splicing a DNA construct into a plasmid. Various techniques are then used to transfect the plasmid into the cell.