Back to Basics - What actually is differentiation?

First post so be gentle with me :)

OK. What actually is differentiation?

Here's the reason I'm asking this.

Is it working out the gradient of the tangent of a curve as per my understanding?

If so, what does it mean to have a tangent (dy/dx solution) which is a quadratic or a polynomial rather than a straight line?

Also, what is the second derivative actually telling you?

What about third derivates and beyond?

Any answers to these would be gratefully received.:)