What is larger -- the set of all positive even numbers, or the set of all positive integers? We will call the set of all positive even numbers E and the set of all positive integers N.
At first glance, it seems obvious that E is smaller than N, because for E is basically N with half of its terms taken out.  The size of E is the size of N divided by two.
You could see this as, for every item in E, two items in N could be matched (the item x and x-1).  This implies that N is twice as large as E
On second glance though, it seems less obvious.  Each item in N could be mapped with one item in E (the item x*2).
Which is larger, then?  Or are they both equal in size?  Why?
(My background in Set theory is quite extremely scant)

They are both the same size, the size being 'countable infinity' or 'aleph-null'. The reasoning behind it is exactly that which you have already identified - you can assign each item in E to a single value in N. This is true for the Natural numbers, the Integers, the Rationals but not the Reals (see the Diagonal Slash argument for details on this result).
-- Added explanation from comment --
The first reasoning is invalid because the cardinality of infinite sets doesn't follow 'normal' multiplication rules. If you multiply a set with cardinality of aleph-0 by 2, you still have aleph-0. The same is true if you divide it, add to it, subtract from it by any finite amount. 
