variable referenced before assignment

[nilamo]

If number is less than 0 (or blank), estim won't be defined, so calling it would be an error.  I'm not sure why you're defining it anyway, just use a while loop directly:

def main():
    number = int(input("Insert the number:"))
    if number == "":
        print("Goodbye!")
    elif number < 0:
        print("Error: negative number")
    else:
        estimation = 1
        while True:
            estimation = (estimation + number / estimation) / 2
            tollerance = 0.000000001
            difference = abs(number - estimation **2)
            if difference <= tollerance:
                break
        number = estimation
    return number