Definition from the AMS Glossary of Meteorology

 normal—1. Referring to a normal distribution. 2. Regular or typical in the sense of lying within the limits of common occurrence, but sometimes denoting a unique value, as a measure of central tendency.

Either sense presupposes a stable probability distribution. 3. As usually used in meteorology, the average value of a meteorological element over any fixed period of years that is recognized as a standard for the country and element concerned.

Often erroneously interpreted by the general public as meaning the weather patterns that one should expect. In the broadest sense, “normals” should consist of a suite of descriptive statistics, including measures of central tendency (e.g., mean, median), range (e.g., standard deviation, interquartile range, extremes), variation, and frequency of occurrence. At the International Meteorological Conference at Warsaw in 1935, the years 1901–30 were selected as the international standard period for normals. Recommended international usage is to recalculate the normals at the end of every decade using the preceding 30 years. This practice is used to take account of the slow changes in climate and to add more recently established stations to the network with observed normals. Normals should be based on actual observations if available; otherwise a recognized method should be used to “reduce” shorter series to the normal period by comparison with neighboring stations. Recognized methods of adjusting for inhomogeneities should be used to account for breaks or gradual changes introduced into the data record by changes in the hours of observation, in the observational practices, in the site or instruments used, or by a gradual change in the character of the surrounding country, such as the growth of a city. The years covered by a normal should always be clearly stated, since averages for different periods of the same length are rarely the same. See climatological standard normals.

This is a very long and detailed definition.  Typically, when we talk about “normal” in meteorology, we are talking about the second part of it, and usually it is in the sense of climate.  Many people don’t realize that “normal” in regards to climatology is a 30 year average, and that average can change at the beginning of every decade.  So when you look at the almanac data for a particular day in a specific city and you see the “normal” or “average” high temperature for that day, you are looking at an average taken from 30 years of data for that day.

This fact should have implications for how we understand daily temperatures and how they fall in line, or don’t fall in line, with what is “normal.”  As time moves on, what is “normal” changes at least one decade at a time.  So really “normal” is a relative term.