The British Society for Immunology has a nice and understandable explaination of vaccines here: https://www.immunology.org/celebrate-vaccines/public-engagement/guide-childhood-vaccinations/how-vaccines-work
If you want a slightly more scientific version, take a look at: Clem (2011) Fundamentals of Vaccine Immunology, available at https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3068582/