What type of immunity is a vaccine?

It depends on the vaccine. Capsular vaccines like pneumonia, H. influenza, and meningitis cause an increase in antibodies by B cells. Some also use helper T-cells to increase memory by conjugating a protein to the capsule (which is made out of sugars). Live vaccines such as chicken pox and the nasal influenza vaccine recruit the entire immune system include T-cells (including killer T-cells), B-cells, and hence antibodies. Live vaccines also tend to last more robust and long-lasting immunity.