calculus - What exactly is a differential?

I've seen the formula for differentials alot, namely

$$dy=f'(x)dx$$

but what I think when I see this is that someone is manipulating the "formula"

$$f'(x)=\frac{dy}{dx}$$

When I think of "differential equation", I think of

$$f\left(x,y,\frac{dy}{dx},\frac{d^2y}{dx^2},\cdots,\frac{d^ny}{dx^n}\right)=0$$

not

$$f(x,y,dy,dx,\cdots)=0$$

I've heard that $\Delta y$ and $\Delta x$ can be approximated by $dy$ and $dx$ (or maybe its the other way around?), but that doesn't make much sense to me. If you replace $dy$ and $dx$ by $\Delta y$ and $\Delta x$, you sort of have Euler's method, but this still doesn't clear much up for me. So,

What exactly is a differential, and why is it useful?