Algorithms, I saw other methods in the commets, but I like this one:

The square root of a number like 9 is 3, 9 is called the radicand. Now if you have x^½ = y then x/y = y. So the radicand decided by the results is the result. 9/3 = 3.

So let's try the following for say 16.

Guess: for example 6.

16/6 = 2.6667

Lets take the average of your guess and this number:

(2.6667+6)/2 = 4.3334

And try this number:

16/4.3334 = 3.692

Take the average:

(3.692+4.3334)/2 = 4.01

16/4.01 = 3.99

(4.01+3.99)/2 = 4

16/4 = 4

So 4²=16, the square root of 16 is 4. And you find a number that is pretty close really quickly.