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the mean and the standard deviation
A "bell" shape.
The normal distribution has two parameters, the mean and the standard deviation Once we know these parameters, we know everything we need to know about a particular normal distribution. This is a very nice feature for a distribution to have. Also, the mean, median and mode are all the same in the normal distribution. Also, the normal distribution is important in the central limit theorem. These and many other facts make the normal distribution a nice distribution to have in statistics.
The standard normal distribution or the Gaussian distribution with mean 0 and variance 1.
The distributions can have any shape that you like.