For a given real number a\geq 0, the symbol \sqrt{a} represents the unique non-negative real number whose square is a. That is, (\sqrt{a})^{2}=a.
Since 3^{2}=9 it follows that \sqrt{9}=3.
Proving that \sqrt{a} exists and is unique in the general situation mentioned above from the foundations of arithmetic is actually no easy feat. Take real analysis someday if you want to learn more.