Set the upper and lower limits. For example, if you want to approximate the square root of 5, you could start with upper and lower limits of 0 and 100. The square root of 5 is clearly greater than 0 and clearly less than 100, but it would save a few steps if you choose more reasonable limits like two and three. Two is a lower limit because 2 X 2 = 4, and three is an upper limit because 3 X 3 = 9. The square root of 5 must be between 2 and 3.
Choose the number half way between the limits for the current candidate. Square the candidate and compare the results to the number you are approximating. If it is low, it becomes the new lower limit. If it is high, it becomes the new upper limit. For example, if you are approximating the square root of 5 and starting with the limits 2 and 4, the first candidate is 2.5. The candidate squared is 2.5 X 2.5 = 6.25, which is high because 6.25 is greater than 5. This means that the candidate 2.5 is the new upper limit. After one step of the algorithm the lower limit is 2.0 and the upper limit is 2.5.
Continue the algorithm until the error is as small as you need it to be. For example, if you need the error to be less than 0.00001, continue the algorithm until the upper limit minus the lower limit is less that 0.00001. The candidate at that stage is between the upper and lower limits and so is the square root you are approximating. This means that the candidate is within 0,00001 of the square root.