There is a brief description of approximating square roots in this earlier question and there is a link there to a more precise explanation. In fact, there are quite a few technical methods of doing square roots. In our Math Central Resource Room, there is
this article on a a geometric view of
a classical algorithm used in years gone by.
There are many other algorithms described on the MathWorld site here, including one used by Newton and one that uses binary numbers.
For a really simple, but rather slow method using long division, you just guess. For example, I guess the square root of 1029 is 30. Now I divide my guess into 1029 and I get 34.3. What I want is for my quotient to be as close as possible to the guess. So I'll improve my guess by choosing a number between my last guess (30) and my quotient (34.3). I'll try 32: Divide 32 into 1029 and you get 32.15. Now I'll guess something between those: how about 32.1...and so on.
Hope this helps!
Penny and Sue