Can anybody explain to me what characteristics make a distribution a normal distribution?

The only one I am sure about is that a Normal Distribution is symmetrical.

-What about the 68,95,99.7 rule in which 68% of the distribution fall under the first standard deviation, 95% fall into the second, and 99.7% fall into the third? Is this a necessary property of a Normal Distribution?

Finally, this is what I am primarily uncertain about. Some distributions show signs of kurtosis in which a lot of values end up around the mean or are at the tail. Are these distributions also Normal Distributions? And does the 68, 95, 99.7 rule apply to these distributions also. In other words, if a distribution is highly peaked in which many of the values fall around the mean, does the first standard deviation still encompass 68% of the distribution?