Your main Navigation Menu has no sub items under any of those links such as Shop.
You have a second menu which is the Mobile one in the bar which has sub navigation links which seem to work fine in firefox.
If you intended this to be under one of the main navigation links above you need to add it as a child of one of those menu items.
Another thing I noted is you have your Search form HTML in there twice sitting on top of each other.
If you look at the second menu, when you put your mouse over Apple, everything followingit opens.
It should open one by one when you put your mouse over it. This is where I am running into problems.
I'll add more items to it so you can see it better. I deleted my original menu thinking if i redid it, it might work correctly.
Also I appreciate you letting me know about the search form being there twice.
Ok, with you.
You need to polish up more of your CSS further.
Because you have the hover and you have a ul inside the li it is treating that and the second ul li inside it.
There is a zip file here: http://forums.adobe.com/docs/DOC-1903 Which is of a menu system, pure css I did. I think if you download that and look it has sub menu's etc you can have a look at the CSS and see how it is all working. It was built with how BC outputs menu's in mind so you should find it useful.