Essentially you start out with two numbers you want to find the GCD for and divide the larger by the smaller 488/90 which gives 5*90 and a remainder of 38.
You then take the smaller number of the previous step and divide it by the remainder so 90/38 which gives 2*38 and a remainder of 14
Repeating the process you get 38/14 which is 2*14 with a remainder of 10.
14/10 which is 1*10 with 4 remainder
10/4 which is 2*4 with 2 remainder
and with the last step you get 4/2 which is 2*2 with 0 as the remainder.
Whenever you reach a 0 remainder the last nonzero remainder is the greatest common divisor which in this case is 2.
So dividing 488/2 and 90/2 you get 244/45 as the simplest form.
Checking the result with wolframalpha shows that it's correct http://www.wolframalpha.com/input/?i=488...