The central limit theorem asserts that means of independent, identically distributed variables will converge to a normal distribution provided they are light tailed enough. This means that even when the exact distribution is not known for some quantity, if there is some form of averaging process going on you will eventually end up with normal distributions. This is the basis for a long list of statistical decision procedures, and since we are likely to end up with a normal distribution for many cases, we also want many variations of normal variables.