This is an interesting question to examine. There are several different meanings in using this statistical term.
Conversational Approach. Very often people refer to climate normal to assess the "usual or typical" climate state at an area or a location. "Usual" or "Typical" in the language of statistics means the most frequently occurring.
Technical Approach. To complicate the story, the term "normal" implies usage of a Normal distribution properties. In case of Climate Variable Normal, it means a Normal Distribution was fit to the climate data.
NWS Directive Approach. Use the NWS Directives 10-1004 on Climate Records http://www.weather.gov/directives/sym/pd01010004curr.pdf to get the definition.
Question for you to think about: How would you explain what the climate normal is to a curious and statistically-savvy customer at your local office?
Exercise for you to do:
1. Obtain your local office area climate normal using xmACIS, located at http://xmacis.nrcc.cornell.edu/*, using the drop down menus to select the Daily/Monthly Normals or Temperature/Precipitation Graphs.
*NOTE: To access this site you have to use your NOAA computer. If you are tele-working today use http://www.cdc.noaa.gov/data/usstations/
2. Share with the group what your climate normal is?
3. Be prepared to discuss your case.