Why can't you divide by 0?

Division by zero is an operation for which you cannot find an answer, so it is disallowed. You can understand why if you think about how division and multiplication are related.

12 divided by 6 is 2 because 6 times 2 is 12
12 divided by 0 is x would mean that 0 times x = 12
But no value would work for x because 0 times any number is 0. So division by zero doesn't work.

Another explanation
Let's look at some examples of dividing other numbers.
10/2 = 5 This means that if you had ten blocks, you could separate them into five groups of two.
9/3 = 3 This means that if you had nine blocks, you could separate them into three groups of three.
5/1 = 5 Five blocks could be separated into five groups of one.
5/0 = ? Into how many groups of zero could you separate five blocks?

It doesn't matter how many groups of zero you have, because they would never add up to five since 0+0+0+0+0+0 = 0. You could even have one million groups of zero blocks, and they would still add up to zero. So, it doesn't make sense to divide by zero since there is not a good answer.

If you know a little bit about multiplication, you could look at it this way:
10/2 = 5 This means that 5 x 2 = 10
9/3 = 3 This means that 3 x 3 = 9
5/1 = 5 This means that 5 x 1 = 5
5/0 = ? This would mean that the answer x 0 = 5, but
anything times 0 is always zero.
So there isn't an answer.
Another explanation

For one thing, when you divide one number by another, you expect the result to be another number. Look at the sequence of numbers 1/(1/2), 1/(1/3), 1/(1/4), ... . Notice that the bottoms of the fractions are 1/2, 1/3, 1/4, ..., and that they're going to zero. If there's a limit to this sequence, we would take that number and call it 1/0, so let's see if there is. Well, the sequence turns out to be 2, 3, 4, ..., and that goes to infinity. Since infinity isn't a real number, we don't assign any value to 1/0. We just say it's undefined. But let's say we did assign a value. Let's say that infinity is a real number, and 1/0 is infinity. Then look at the sequence 1/(-1/2), 1/(-1/3), 1/(-1/4), ..., and notice again that the denominators -1/2, -1/3, -1/4, ..., are going to zero. So again, we would want the limit of this sequence to be 1/0. But looking at the sequence, it simplifies to -2, -3, -4, ..., and it goes to negative infinity. So which would we assign to 1/0? Negative infinity or positive infinity? we say that infinity isn't a number, and that 1/0 is undefined.

Another way
Here's a little experiment for you to try on your calculator. Observe the output when you try the following set of calculations:
1/1
1/.1
1/.01
1/.001
1/.0001
etc...

until your calculator can't go any further or you get tired. You should notice that the answers continue getting larger and larger.
Another way of thinking of it is to imagine filling a box with apples. Say a box can hold 100 apples. Now try filling it with apples that are half the size of these apples. You can put 200 in the box. Now imagine a special, magic apple that takes up no room at all. How many can you put in the box?
Well, the answer is... there is no answer! That is why mathematicians refer to numbers that are divided by 0 as "undefined." Some people tend to think of them as being infinite, but this isn't exactly true. There simply is no answer.