Vaccines are made from the same germs (or parts of them) that cause disease. But the germs in vaccines are either killed or weakened so they won't make you sick. There is no need to worry about catching a disease from the vaccine.
Like any medication, vaccines can cause side effects. For the most part, these are minor and go away within a few days — for example, a sore arm or low-grade fever. Serious reactions to vaccines are extremely rare. Compared with the dangers of the diseases they can prevent, vaccines pose a very, very small safety risk.
There is a great deal of confusing information about vaccine safety. The United States currently has the safest, most effective vaccine supply in history. Before a vaccine can be used, it must go through years and even decades of testing. Once a vaccine is in use, the Centers for Disease Control and Prevention (CDC) continues to monitor its safety and effectiveness, year after year.
Want to learn more? The CDC has a lot of great information about vaccine safety here.