What is a-b?
Spend some time thinking about it. Now, how do you explain a-b to someone who knows only how to add? You'll have to explain it only in terms of 'addition', right? How do you do that? Think...
You say, a-b is "that number, which when added to b, gives a". That is, to find a-b, you solve the following problem:
"Find c such that b+c = a"
Now switch to this question:
What is a/b?
Again, spend some time thinking about it. But you won't think much, because by now, you know exactly what I mean! How do you explain a/b to someone only in terms of 'multiplication'? How do you do that?
You say, a/b is "that number, which when multiplied to b, gives a". That is, to find a/b, you solve the following problem:
"Find c such that b x c = a"
Let's now answer the question:
What is 0/0?
Using the recently established definition of division, how do we explain 0/0 in terms of multiplication? We say, 0/0 is "that number, which when multiplied to 0, gives 0".
That is, to find 0/0, we have to "find c such that 0 x c = 0". What do you think the answer should be?
You're right, any number!
Even our query was wrong. It should have been: "What are those numbers, which when multiplied to 0, give 0"?
So, 0/0 is not really one number, it represents many numbers. Pick a number, and that's 0/0. It's like a blank cheque on which you can enter any amount. But how can you have multiple values for a mathematical constant (which 0/0 is supposed to be)? Is it allowed? It becomes a constant variable! Paradoxical, right?
Now, same is not the case with other mathematical constants, like 1, -19, 25000, 3.93, 3+2i, etc. Each number has a unique value. That is why they are called "determinate", since they have single, unambiguous, deterministic values.
But 0/0 doesn't have such a value. It could be anything; it cannot be determined. It's "indeterminate". And such an entity is dangerous for maths; and we do exactly what we do with dangerous people in the society:
So, 0/0 is disallowed in maths.
Image credit:
http://www.123rf.com/photo_16105671_bad-guy-in-jail.html
EDIT 1:
Someone below was curious about the difference between 0/0 and infinity (1/0). To understand the difference, just try to apply that definition of division to 1/0. Find a number that when multiplied to 0 gives 1. You'd be amazed to find that there exists no such number!
So, 0/0 exists, but has too many values (all values), but 1/0 doesn't exist at all. It's only an abstraction. As far as maths is concerned, both are dead! You can say that 0/0 dies of overeating, and 1/0 dies of starvation!
EDIT 2:
Someone else in the comments suggested the "divide a into b parts" definition. This explanation is only true when b is a positive nonzero integer, i.e., when b lies in {1,2,3,...}.
It has no meaning for many kinds of b's. Can you:
divide 4 into -2 parts?
Or 8 into 0.5 parts?
Or 6 into 4/3 parts?
Or 2 into 2+3i parts?
Or 9 into √2 parts?
None of these statements make sense. You have to see beyond that trivial definition of "divide a into b parts". In the same way, dividing into 0 parts doesn't make sense. It's nothing special.
It was also suggested that 0/0 is imaginative. That's right. 0/0 is a concept, not a number. It's a concept that's been disallowed for use in formal mathematics. Maths is full of rules. For example, can you tell me what is the value of "7+%4!-6/*"? No one can, because it's nonsensical, and is disallowed in maths.