calculus - What's the meaning of a differential (as opposed to a derivative)?

Say $y=f(x)$. To my understanding, the meaning of the derivative $\frac {dy} {dx} = f'(x)$ in English would be something like "the change in y with respect to the change in x at a given point is $f'(x)$."

After a symbolic manipulation of the derivative where we "multiply" both sides by $dx$, we get the differential $dy=f'(x)dx$. My question is, what does the differential actually mean (verbally), and how is it different from the derivative?