Vaccine reactions, or "vaccine side effects," can be classified as local, systemic, or allergic, generally increasing in severity respectively.

Examples of local reactions include pain, swelling, or redness at the site of the reaction. These most commonly occur with inactivated vaccines like DTaP. One specific type of local reaction is named the Arthus reaction, and is the result of an antibody response to the injection. This requires that antibodies had already been produced during prior exposure to the vaccine and is mainly seen in toxoid vaccines like Diptheria and Tetanus.

Systemic reactions include many of the symptoms associated with a systemic immune response, including fever, malaise, myalgia, headache and loss of apetite. Though possible with any whole cell vaccine, they are most common following live attenuated vaccines such as a nasally administered influenza vaccine.

Allergic reactions to a vaccine can be to the antigen itself, or any other ingredient within the vaccine. Though rare, they can be life-threatening in severity.

Interestingly, a "vaccine event" is distingished from a vaccine reaction in that it is any event following a vaccination, whether a true adverse reaction or a coincidental occurance. For instance, guidelines for instating permanent DTaP precautions in a pediatric patient include:

- Temperature of 105°F or higher within 48 hours of a dose of DTaP
- Collapse or shock-like state within 48 hours of a dose of DTaP
- Persistent inconsolable crying lasting 3 or more hours occurring within 48 hours of a dose of DTaP
- Seizure, with or without fever, occurring within 3 days of a dose of DTaP.

Sources: 

GeneralRulesImmunizationCDC

"Immunization - You call the shots" from CDC.gov