How does a vaccine lead to the production of antibodies and why does this provide immunity?

Vaccines are designed so that they contain antigens - for example viral antigens from the virus we wish to protect against. When these are injected they are recognised as foreign by white blood cells called Antigen presenting cells (e.g. Dendritic cells). These engulf the antigens and present them to T helper cells. T helper cells then release substances which stimulate B cells to divide into plasma and memory cells. Plasma B cells produce molecules called antibodies. These antibodies function in a number of ways to neutralise the virus - for example by binding to the viral antigens and clumping the virus together. This process of vaccination causes a heightened response if the same antigen is to re-enter. The memory B cells remain even after the initial vaccination/infection has occurred, and can respond rapidly by differentiating into plasma B cells and producing antibodies.