Set up the problem. Draw a radical sign that has plenty of room. Write the number you want to take the square root of under the radical. Starting at the decimal point, write the digits of the number in pairs. Sometimes this will mean writing the leftmost digit as a single number. For example, if you were taking the square root of 54321 you would group the digits like this: (5 43 21. 00 00 00)^0.5. When you finish the algorithm, there will be one digit of answer for each of these groups.
Look at the leftmost group, and find the largest number such that the square of the number is equal to or smaller that the leftmost group. Write this number on top or the radical -- right over the leftmost group. From now on this number will be called "the answer." Square the answer and write that under the leftmost group. Draw a line (this is where it starts looking like long division). Subtract the square from the leftmost group and write that under the line. Bring down the next group of two and put this under the line. The number under the line will be called "the difference."
Multiply the answer by 20 to get "the first factor." The first factor is written outside tha radical to the left of the difference. Now you need to find a single digit N such that N X (First factor + N) is the largest number that is less than or equal to the difference. This number will be called "the final factor." Write the final factor under the first factor, then write N X (first factor + N) next to the final factor and under the difference. Draw a line and subtract to find the next difference. Write N above the radical and over the last group of two that was brought down. Repeat this step until you have the accuracy you want.