You need to find the factors of the number you're dividing. If it's an even number you can divide by 2, so do this and note the factors as you find them. Therefore you have so far 2 and another number, we'll call n. Look at n, and if it's even you have another 2, so you would write 2*2 and another n. When you no longer have an even number, or if the first number was odd, you check to see if 3 goes into it. You make a note and divide the number by 3 so you now have ...3 and another n. You may even have 2*2...*3*n. Again you try 3, until you can't divide exactly, then you move on to 5 (numbers ending in 5 or 0 are exactly divisible by 5). You carry on like this noting what factors you've found as you go along. The numbers you pick as divisors will always be prime: in order here are the first few: 2, 3, 5, 7, 11, 13. A prime number is one which can't be divided exactly by a number smaller than it (actually up to its square root). 4 isn't prime because 2 goes into it; 9 isn't prime because 3 goes into it and all primes apart from 2 are odd numbers.
Now's a good time to look at examples. First, 660.
It's even so 2 goes into it. So we have 2*330. 330 is also even: 2*2*165. 165 is divisible by 3, so we divide by 3 and we have 2*2*3*55. 55 ends in 5 and it's not divisible by 3, so we divide by 5 and get 11, and now we have 2*2*3*5*11. 11 is prime so we stop. 660=2*2*3*5*11. This breakdown means we can spot the factors other than prime factors. 2*2 means that 4 is a factor; 2*3 means that 6 is a factor; 3*5 means that 15 is a factor; 2*5*11=110 means that 110 is a factor, and so on. So if you needed to divide 660 by 15 you cross out the factors that make 15 (3 and 5) leaving 2*2*11=44.
Let's try another example: 2030.
2030=2*1015=2*5*203=2*5*7*29. 29 is prime.
How do we know how far to divide? Take 9042=2*4521=2*3*11*137. Is 137 a prime number? We only have to try and divide by numbers up to the square root of 137. What does that mean? The square root is the number which when multiplied by itself is closest to the number we're trying to divide, but smaller than it. 11*11=121; 12*12=144, so we only have to go up to dividing by 11, because 11*11 is less than 137 while 12*12 is greater, and the square of the next prime is 13*13=169, which is bigger than 137; so we only need to go up to primes up to 11, which doesn't divide exactly into 137, so 137 must be a prime number.
Now try some numbers yourself!