How do square roots relate to every day life?
Due to the nature of this question, I've strayed a bit into opinion. Should someone wish to correct or replace it, feel free.
The square root is an integral part of the world we live in. It's one of the fundamental functions of math used in many kinds of complex calculations. Computers, as well as everything like computers (e.g. phones), use square roots for many of their functions. Aviation makes use of square roots when calculating the speed required for hydroplaning. These are, of course, things which the average person will benefit from, but will not have to interact with. The average person in their everyday life will not use square roots despite what their teachers may say. That said, it is important to learn about how to find a square root and what it really is. Having a better understanding of math than you might think you need will make many things much easier later in life. While you might not use square roots, learning about them will help you understand other mathematical functions which will be used (namely, in taxes). I know a huge number of people who can't even find what sales tax they would need to pay if the computer didn't tell them.
Even without that, though, if you enter into an occupation which makes use of math (e.g. engineering, computer sciences/programming, ANY science field), then square roots will definitely be used in many of the equations you will likely be working with. This doesn't apply to everyone, but it applies to anyone who graduates from college and enters a field of science.
Studying marine life (the way a school of fish move, specifically) and studying the best arrangement for wind turbines use the same equation, a very complex one which uses square roots. Social sciences use the square root of a population to determine the proper number of representatives in a governing body (this is only one method, but it's used in other places as well).
So, even if a subject doesn't seem like it would use a square root, that's no guarantee it won't.