Hm, well, you've got 20 balls total, and 13 are metal. So when you choose one at random, you've got a 13/20 chance of choosing metal. Compared to the 7/20 chance of choosing wood, it's about twice as likely (13 is about 14, which is 2*7) that you'll pull a metal ball than you would a wooden one, which disagrees with what you're being told in the setup. You're 3 times likelier to choose wood than metal, which means that there's some hidden factor that makes metal much less likely than it otherwise would be. You can't actually come to a good conclusion about how this is going to influence the rest of the problem, but the simplest assumption is that there's a single factor skewing your odds of getting metal downward, I think. To figure this out, you can compare your actual chance of drawing metal to your expected chance - in this case, 13/20 vs 5/20. So you're 5/13 times as likely to draw metal as you'd expect to be from the numbers alone.
Now, you've correctly figured out that you've got a 1/4 chance of drawing metal the first time, based on what you've been told about the actual odds. If you put the ball back before drawing a second one, your odds of drawing metal the second time are the same, so your odds of drawing metal both times are (1/4)*(1/4) = 1/16. In this case, telling you how many balls of each kind there were was a red herring meant to confuse you, which is something many teachers are fond of.
If you don't put the ball back, or you draw two balls at the same time (I believe these are equivalent, a better statistics person than I could confirm it), you have one fewer metal ball that you could draw. So, now, your expected odds are 12/19, but there's that mysterious voodoo factor that you have to account for. Since you've got no reason to assume it should change, I believe that your chance on the second draw of drawing metal is (12/19)*(5/13) = 60/247. Multiplied by your 1/4 chance of drawing metal the first time, your final chance is 15/247.
EDIT: We have a ninja, I'm gonna read his post. If we disagree, trust him over me.
EDIT: And I'm completely lost as to that explanation. I defer to a wiser man, but I cannot see what marking individual balls adds to the problem. It's probably a more sensible strategy than my arbitrary factor, but I'll be damned if I can follow the logic.