There are a number of conditions that can make a data set not "normal". Among these are:
(1) The data could be bimodal or multimodal. For example if you look at the average height of humans, you will find two "peaks"; one for females and one for males.
(2) The data could be positively or negatively skewed. If you graphed income level versus number of people achieving that income, the graph would be negatively skewed as far more people earn little or nothing compared to the very wealthy.
(3) The data could be constant. The distribution of digits in a truly random string of numbers has the same percentage of 1's as 7's, etc...