They are used extensively in physics and engineering, as well, of course, by mathematicians. That's the easy part of the answer. Much more complicated is explaining what they are.

It's hard to understand what a Differential Equation is without first knowing a bit of Calculus. If you're asking about them in the first place, maybe you know what a derivative is already, but here's a brief explanation:

Suppose a function f(x). f'(a) (pronounced f prime of a) is the instantaneous rate of change of the function at the point a. f'(x) is the general rate of change at any point x in the function.

Consider f(x) = x^2, and let's say a = 3.
Then f(a) = 9, but what is the rate of change at point a in the function? Well, you're going to have to trust me here if you don't know derivatives already that f'(x) = 2x.
So f'(a) = 18. The rate of change at the point 3 of f(x) = x^2 is 18, and the general rate of change at any point x is 2x.

Okay, now for the really confusing stuff.

You can take a derivative of a derivative. So in the example above, f''(x) would be called the Second Order derivative of f(x). Again, take my word for it that it is 2. f'''(x) which is the Third Order derivative would be 0. But all you really need to take from this is that there are derivatives of increasing order, as you keep taking the derivative (called differentiating--that word seems familiar.) of a function over and over.

So what's a differential equation? Well, it's a function ultimately, so let's keep calling it f(x). But this function is expressed by the derivatives of various orders of another function. So we have our f(x), but if it's a differential equation, it'll look something like f(x) = f'(y) + f''(y) + f'''(y)... and so on.

And it just keeps getting more confusing from there, so I'll stop. But that's the basic idea. I hope it helps.
Source(s):
http://en.wikipedia.org/wiki/Differential_equation

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