The Gaussian distribution plays a central role in statistics because of a mathematical relationship known as the Central Limit Theorem. To understand this theorem, follow this imaginary experiment:
1.Create a population with a known distribution (which does not have to be Gaussian).
2.Randomly pick many samples of equal size from that population. Tabulate the means of these samples.
3.Draw a histogram of the frequency distribution of the means.
The central limit theorem says that if your samples are large enough, the distribution of means will follow a Gaussian distribution even if the population is not Gaussian. Since most statistical tests (such as the t test and ANOVA) are concerned only with differences between means, the Central Limit Theorem lets these tests work well even when the populations are not Gaussian. For this to be valid, the samples have to be reasonably large. How large is that? It depends on how far the population distribution differs from a Gaussian distribution. Assuming the population doesn't have a really unusual distribution, a sample size of 10 or so is generally enough to invoke the Central Limit Theorem.
To learn more about why the ideal Gaussian distribution is so useful, read about the Central Limit Theorem in any statistics text.