Short answer:
It's used to make video games.

Longer answer:
When you go about programing any sort of AI for a computer game you'll need to use square root to determine the distance of any two given points on the screen.

Engineering :-)
For example: http://wtnasa.com/

Actually it's exactly the opposite :-) Square root of a given number is an exact value. Only if you try to calculate it and represent as a decimal number you get an approximation.

As for why it exists in the first place - if you like a more philosophical answer, it's a link between stuff in two dimensions (surface of a square) and stuff in one dimension (side of a square). If you want to go from three dimensions to one (cube volume to cube height) you use a cube root, and so on :-)