Quick definition: In a ring R, a zero divisor is a nonzero element r in R such that for another nonzero s in R, rs = 0. A ring without zero divisors is an integral domain, sometimes also known as an entire ring.
Remember the zeroproduct property? Your algebra teacher might have stressed it back in the day. It says that the only way to multiply two numbers and get zero is if one of the two numbers is zero to begin with. It might not have seemed like much back then. It's a little surprising to see how much in fact depends on that property.
A zero divisor is something that violates the zeroproduct property. Of course, no pair of ordinary nonzero numbers can be multiplied together to get zero, so we'll have to look for zero divisors in more exotic realms. Hence consider a different set of things that can be added and multiplied together. This is called a ring. Here's an example of such a ring: the integers modulo 6, denoted Z/6. It has six elements, {0, 1, 2, 3, 4, 5}, and we just loop around back to 0 if we have to go over 5. Thus, for example, 5+2 = 7 = 1 (mod 6) and 3·5 = 15 = 3 (mod 6). More importantly, Z/6 has a pair of zero divisors, 2·3 = 6 = 0 (mod 6).
Zero divisors are icky and generally undesirable. We like rings without zero divisors, rings such as the integers; zero, one, two, three, and their negatives. In fact, we like them so much, that we have a special name for them, integral domains. Serge Lang also has modestly succeeded in spreading the term entire ring for a ring without zero divisors, due to some linguistic analogy with his native French.
There are several things that go wrong when a ring has zero divisors. A sample:

The cancellation law fails.

Zero divisors cannot have a multiplicative inverse. Thus a ring with zero divisors has no hope of being a field such as the rational numbers.

The degrees of polynomials over rings with zero divisors doesn't necessarily add up when you multiply the polynomials.

A polynomial of degree n over a ring with zero divisors may have more than n roots.
Allow me to explain each in turn.
In an integral domain, the cancellation law says that if we ever have an equation like ab = ac, and a is not zero, then we can cancel the a and conclude that b = c. The proof depends on the zeroproduct property:
ab = ac,
ab  ac = 0,
a(bc) = 0,
but a isn't zero, and by the zeroproduct property, it follows that bc = 0, and b = c. On the other paw, in the ring with zero divisors Z/6, 2·1 = 2·4 (mod 6), but 1 is not equal to 4 modulo 6.
For our next act, we will have to assume that we are dealing with a ring with one in it, the multiplicative identity. Take any pair of zero divisors r and s, in other words, neither r nor s is zero and yet rs = 0. Suppose for contradiction that r has a multiplicative inverse, that is, there exists some element t in the ring such that tr = 1. This leads to nonsense, because if we multiply both sides by s we see that
trs = s,
t·0 = s,
0 = s,
since anything times zero is zero in any ring. But we are led to conclude that s is zero, contrary to the assumption that it wasn't. Therefore, not r nor any other zero divisor, could possibly have a multiplicative inverse. To put it another way, dividing by zero divisors is just as bad as dividing by zero. Side effects may include absurdity, poppycock, and insanity. Don't ever let me catch you doing it.
Now for the polynomials. Usually, degrees add up when you multiply. The polynomials over the integers certainly do this. If you take x^{3} + 2 x of degree 3 and multiply it by x^{2} of degree 2, you get x^{5} + 2 x^{2} of degree 3+2=5. But let's look again at our favourite example of a naughty ring, Z/6. Two polynomials over Z/6 are 2x^{2} + 5 x and 3x^{2} both have degree 2, but their product 3x^{2} is only of degree 2, not 4. The problem is that when you multiply the two leading terms of each polynomial, you get zero, since their coefficients are a pair of zero divisors.
That degrees of polynomials fail to add up is a minor annoyance. A more serious problem with polynomials over rings with zero divisors is that they may have more roots than their degree. With ordinary numbers, a very basic theorem is that a polynomial of degree n cannot have more than n roots, and therefore has a unique factorisation. Many important results hinge on this basic one, and not surprisingly, the proof of the basic theorem depends on the zeroproduct property. Thus, sometimes you can factor a polynomial in more than one way over a ring with zero divisors!
For instance, over Z/6, consider the seconddegree polynomial x^{2} + 5x. It has four roots: 0, 1, 3, 4, and two possible factorisations: x(x + 5) and (x + 2)(x + 3). Notice how our devilish duo make another appearance: one possible factorisation of the polynomial has the pair {2, 3} of zero divisors for roots.
Zero divisors are an unpleasant but necessary fact of algebra. Sometimes, it is impossible to avoid them, as many respectable rings have zero divisors. The ring of matrices over any ring comes to mind as an example of a reputable ring with zero divisors. I therefore cannot ask you to never go near a zero divisor. On the other hand, I urge you to exercise caution when working with rings with zero divisors and to not make rash assumptions. Above all, avoid nonsense!