For many, the square root is one of the first mathematical concepts they will learn outside of simple addition, subtraction, multiplication, and division. The square root is a concept that not only combines multiplication and division, but also is used extensively in complex formulas, concepts, and equations later on.
At its most basic, the square root is a number which, multiplied by itself, gets you an original number. A good example is the number 4. If you take the square root of 4, you are left with 2. Thinking of it another way, you need to ask yourself what number could be squared to equal the number under the radical sign. The radical sign is what indicates that a number or variable is having its square root taken, and it looks like (square root of) x, or in exponent notation, like x (x to the one-half), where x is the original number.
All non-negative real numbers have a non-negative square root, including non-whole numbers or irrational numbers. For example, the square root of 45 is 6.7082038325. It's not an attractive number, but if you square it, you'll still get 45. There are also a whole set of numbers called imaginary numbers, of which the square root plays an important role.
Imaginary, or complex, numbers exist because if you take the square of a positive or negative number, you'll always end up with a positive solution. To take the square root of a negative number, there would have to be a pair of numbers that could multiply together to equal it. To represent this, there is a variable i, where i = (square root of) -1, and equals -1 when squared. This allows people to deal with imaginary numbers.
The square root is also an important part of solving algebraic equations. To solve an algebraic equation, you have to keep the two sides equal, so if a variable is squared, you can take the square root on both sides of the equation to isolate the variable and determine its value. The square root is used in a lot of helpful formulas too, like the quadratic formula:
x = -b +/- (square root of) (b2 4ac) divided by 2a
As you get higher up in mathematics, such as in calculus or statistics, the square root will be one of the most important tools you can use in solving problems. From the distance formula to quadratic equations to derivates and integrals, the square root is a helpful and important little symbol that will serve you well.