Although Gauss was not the first to publish his findings on the normal distribution (which was de Moire in his work with gambling, probability theory, and the binomial distribution) it is accredited to him because he seems to have been working with it long before de Moivre.

The importance of the Gaussian distribution can be felt today. Modern applications of the normal distribution find themselves in many fields where analysis of error is required such as social and physical sciences, risk management, probability, statistics, educational research, and industry.

Why?

The central limit theorem indicates that independent, identically distributed samples from a population will have an approximately normal distribution which is crucial in order to make inferences. If we know the probability distribution of a sample we know what range of values are likely to be the true value of the overall population of interest.

Likewise the method of least squares, which Gauss used in his determination of the planetoid Ceres, is used extensively in Regression analysis, where one uses data to determine a line of best fit for the purposes of seeing how one or more variables affect a specified response variable. The difference between observed and predicted (by our line of best fit) are called residuals and their distribution is normal.