Normal distribution parameters
The family of normal distributions consists of symmetric bell-shaped distributions that are defined by two parameters, µ and σ, the distribution's mean and standard deviation.
Normal distributions as models for data
The sample data rarely gives enough information for us to be sure that the underlying population is normal, but a normal model is often used unless there is obvious non-normality in the data.
Even if the sample data are obviously skew, a normal distribution may be a reasonable model for a nonlinear transformation of the values (e.g. a log transformation).
Distribution of summary statistics
A more important reason for the importance of the normal distribution in statistics is that...
Many summary statistics have normal distributions (at least approximately).
The Central Limit Theorem shows that the mean of a random sample has a distribution that is close to normal when the sample size is moderate or large, irrespective of the shape of the distribution of the individual values. The following are also approximately normal when the sample size is moderate or large...