I'm assuming you're referring to differential calculus. dy/dx is the relationship between the derivative of the function and x.
du/dx is usually used when a chain function is being derived, for example (10x^2+5x)^5 , where we later use the notation u^5. Thus, the derivative would be dy/du = 5u^4.
It is hard to explain it without having a specific example given. Hope it helps anyway, at least a bit. I'm sure other users can go a bit more in-depth :P