Qiaochu_Yuan already answered your question, but because he was pretty technical with his answer, I thought I should try to simplify the point here a bit. The problem with division by zero is that division is essentially defined through multiplication and existence of certain inverse elements. It's an axiom in itself in group theory that there are inverse elements, that is, for each a, there is x such that a*x = 1. Our notation for x here would be 1/a, and it's easy to see why a * 1/a = 1. Division is defined by these inverse elements: a/b is calculated by a * (1/b), where (1/b) is the inverse of b.

But, if you have both multiplication and addition, there is one interesting thing. If we assume addition is the group operation for all numbers(and we use "0" to signify additive neutral element you get from adding together an element and its additive inverse, that is, "a + (-a) = 0"), and we want multiplication to work the way we like it to work(so that a(x + y) = (ax) + (a*y), that is, distributivity hold, something interesting happens.

Now, neutral element 0 is such that x + 0 = x, this is by definition of neutral element. Now watch the magic happen: 0x = (0 + 0)x
= 0x + 0x So 0x = 0x + 0x.

We subtract 0x from both sides, leaving us with 0x = 0.

Doesn't matter what you are multiplying 0 with, you always end up with zero. So, assuming 1 and 0 are not the same number(in zero ring, that's the case, also, 0 = 1 is the only number in the entire zero ring), you can't get a number such that 0*x = 1. Lacking inverse elements, there's no obvious way to define what it would mean to divide by zero. There are special situations where there is a natural way to interpret what it means to divide by zero, in which cases, go for it. However, it's separate from the division defined for other numbers.

And, if you end up dividing by zero because you somewhere assumed that there actually was such a number x that 0*x = 1, well, that's just your own clumsiness.

Also, you can prove 1=2 if you multiply both sides by zero. 1 = 2. Proof: 10 = 20 => 0 = 0. Division and multiplication work in opposite directions, multiplication gets you from not equals to equals, division gets you from equals to not equals.