If you really cant use the sum() function, here is a function to sum the rows, I've written it explicitly to show the steps but it would be worth looking at list comprehensions once you understand what is going on:
def row_sums(square):
    # list to store sums
    output = []
    # go through each row in square
    for row in square:
        # variable to store row total
        total = 0
        # go through each item in row and add to total
        for item in row:
            total += item
        # append the row's total to the output list
        output.append(total)
    # return the output list
    return output
This can then be used as such:
square = [
   [1, 2, 3, 4],
   [5, 6, 7, 8],
   [9, 10, 11, 12],
   [13, 14, 15, 16]
]
row_totals = row_sums(square)
EDIT:
In answer to your comment, I'd do something like this:
def sum_columns(square):
    # as before have a list to store the totals
    output = []
    # assuming your square will have the same row length for each row
    # get the number of columns
    num_of_columns = len(square[0])
    # iterate over the columns
    for i in xrange(0, num_of_columns):
        # store the total for the column (same as before)
        total = 0
        # for each row, get the value for the column and add to the column total
        # (value at index i)
        for row in square:
            total += row[i]
        # append the total to the output list
        output.append(total)
    # return the list of totals
    return output