"Don't do it! It's bad! Very bad! Never divide by zero!" -- Your Teacher
Dramatic, isn't it? You've probably heard something like that from your math teacher before, and they were right. As soon as you divide by zero, or by a more complex expression that is equal to zero, you have broken the rules of mathematics and your answer is now suspect. Even if you divide something by zero, or by a term equivalent to zero, in the middle of solving a problem, the remainder of your work may be invalid as a result.
Let's think about it from a simplistic point of view. What does it mean to multiply? If I have 3 times 2, that could mean three groups of two, for a total of six. Multiplication is repeated addition in that sense. You can multiply by zero because that means you have zero groups, and thus a total of zero.
But what about dividing by zero? If you represents the slices of a pizza into a fraction, then 4/8 means you have four of the eight pieces. What would 4/0 mean? That you have 4 of the zero pieces? Imagine dividing up 6 dollars among 3 people. Each person would have 6/3, or 2 dollars. But now divide 6 dollars by 0 people. How much does each person get? It doesn't make sense, because there aren't any people to divide the money among! That's why division by zero is undefined. When you divide by zero the answer isn't zero, or infinity, or negative infinity. It's undefined. As in... it doesn't make sense.
The folks at mathforum.org actually put together a nice list of some questions and answers about dividing by zero, and some nice ways of visualizing it in your mind. Check it out to convince yourself that division by zero is undefined.