The normal distribution is popular for two reasons:
- It is the most common distribution in nature (as distributions go)
- An enormous number of statistical relationships become clear and tractable if one assumes the normal.
The normal (Gaussian) distribution is to statistics what Newtonian mechanics is to physics.
Sure, nothing in real life exactly matches the Normal. But it is uncanny how many things come close. And this is partly due to the Central Limit Theorem, which says that if you average enough unrelated things, you eventually get the Normal.
Like classical (Newtonian) mechanics in physics, the Normal distribution in statistics is a special world in which the math is straightforward and all the parts fit together in a way that is easy to understand and interpret. It may not exactly match the real world, but it is close enough that this one simplifying assumption allows you to predict lots of things, and the predictions are often pretty reasonable.
The normal is also statistically convenient. It is represented by two parameters which are arguably the most basic statistics there are: the average and the variance (or standard deviation). The average is the most basic statistic there is. And variance is arguably the second most basic (the average of what's left when you take away the average, but to the power of 2).