According to the Centers for Disease Control and Prevention's Vaccine Glossary, an acellular vaccine is "a vaccine containing partial cellular material as opposed to complete cells." The vaccine manufacturer only uses the antigenic part of the virus, meaning only the part of the organism that is responsible for causing disease (capsule, flagella, part of the protein cell wall).
Acellular vaccines generally do not initiate the strongest immune response. In most cases, one is required to receive a "booster" some time after receiving the first vaccine. Acellular vaccines cannot cause disease and are therefore considered safe.