What does the immune system produce to fight future infections with the same type of virus that is introduced in a vaccine?

The immune system, when coming in contact with a new foreign entity, (Introduced by the vaccine), will first digest the foreign entity with white blood cells to determine its molecular makeup, the white blood cell will then proceed to construct specific antibodies t that will cling to the antigens of the invader, this will help the body know what is foreign and what is friendly! now, after you have fought an infection, your body will set aside certain cells called memory cells, these cells can live for weeks to years within the Lymphatic system as dormant cells, however, if the invader were to show itself again, the antigen presence from the invader would trigger the dormant memory cell to become active and mass produce the antibodies specific for that strain of infection. therefore preventing the show of symptoms due to the early warning system now in place.  

Hope this helps! :D