Pfizer and Moderna provided specific information to the FDA that lists the following possible side effects within 7 days of the vaccination:
Local reactions around the vaccination site (mostly mild pain, redness and swelling).
Fever, chills, fatigue, headache, muscle pain, and joint pain.
These side effects are usually mild, and at worst are moderate. These side effects are a sign the vaccine is working, and your body is mounting the immune response it needs to in order to protect you from severe COVID-19 infection. They are also much less severe and last for a shorter period of time than actual COVID-19 infection.
Concerns about developing infertility, Bell's palsy, alterations of the DNA, and autoimmune conditions are unfounded. These are not side effects of the vaccine and have not occurred more frequently than normal among those who have received the vaccine.
As with any medicine, there is a very remote chance of a vaccine causing a severe allergic reaction, other serious injury, or death. No severe side effects have been reported from the clinical trials.