Do the pfizer and moderna covid-19 vaccines work in the same way as other vaccines?

They are different: Other vaccines include either a component of a virus or bacteria, or an actual live but weakened organism. The new Pfizer and Moderna COVID-19 vaccines are different, because they do not contain any component of the virus. Instead, they have a small slice of the genetic code of the virus in the form of messenger RNA (mRNA). The mRNA gives instructions for your cells to manufacture the spike protein of the virus, which enables your immune system to make antibodies against the virus. Antibodies give you immunity, so you will not get sick if you are exposed to COVID-19. Reference: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7685956/

Vaccines: I won't repeat the excellent information provided by Dr. Rutledge. However, to get an idea of the different vaccine mechanism consider reviewing an excellent powerpoint from the World Health Organization (pg 10 & on). REF: https://www.who.int/docs/default-source/coronaviruse/risk-comms-updates/update37-vaccine-development.pdf?sfvrsn=2581e994_6