2S + 3O2 ==> 2SO3
Let's start off with the unbalanced equation and count atoms of each element on both sides of the arrow.
S + O2 ==> SO3
There's 1 sulfur atom on both sides which is good. However, there's 2 oxygen atoms on the left and 3 on the right. So calculate the Least common multiple (LCM) of 2 and 3 which is 6. And adjust the coefficients on both sides to have 6 oxygen atoms on each side, giving:
S + 3O2 ==> 2SO3
Now we have 6 oxygen atoms on each side which is good, but there's only 1 sulfur atom on the left and 2 on the right. Since the LCM of 1 and 2 is 2, all we need to do is adjust the coefficient on the left to get 2 sulfur atoms there. So:
2S + 3O2 ==> 2SO3
We now have 2 sulfur atoms on each side of the equation and 6 oxygen atoms on each side, so the resulting equation is balanced.