They are called attenuated vaccines. Attenuated means weakened, they do this usually with chemicals and then the attenuated viruses are not able to make you sick, but they are strong enough to trigger the immune response to provide immunity.
An attenuated virus. Attenuated simply means weakened. For example, you will see references to this in regard to the flu vaccines. This is also abbreviated when talking about the flu vaccines as LAIV, Live Attenuated Influenza Vaccine.
That is a good description of what a vaccine is, although some other substances also could be described similarly. The weakened or killed virus in a vaccine is non-threatening to most people. It can be a problem for infants under six months old, some children and adults with under-developed immune systems or who have immune system disorders.