Well, if you know any calculus, then you probably seen some like this the notation dy/dx before.
An example of a differential equation is...
dy/dx = y(x)
What that means is that the derivative of the function y(x) equals y(x). So, what's an example of a function that has itself as a derivative? If you guessed e^x, then you're right! In fact, the whole class of ke^x, where k is a constant, satisfies this condition.
Now, what about this next one...
dy/dx = x*y(x)
So, the derivative of y(x) equals itself multiplied by x. What is an example of such a y(x)? Well, if you guess something like e^(x^2/2), then you're right, and again, all of the k*e^(x^2/2) satisfy this condition.
Now, you probably noticed that we're solving y(x), and not for x. When dealing with differential equations we looking for a function y(x) that satisfies the model/differential equation that we constructed.
It might sound simple at first, but it gets really hard quickly. If you thought integrals were hard, watch out for differential equations.
Of course there is a lot more to it than what's above, but that's an idea.
Where can you learn about this stuff? If you know some calculus, then any introduction differential equations textbook should be enough. Try to pick a newer one to avoid different notations, which can get confusing.
These are examples of the kinds of questions that are asked in diff eqs, a vast subject that contains the applications of derivatives to the real world.
*The first diffential equation represents exponential growth, as in a population or a bank account. The second differential equation models a sinusoidal oscillator, for example a cork bobbing in the water or a pendulum.