Question: So what exactly is a derivative? Is that the EQUATION of the line tangent to any point on a curve? So there are 2 equations? One for the actual curve, the other for the line tangent to some point on the curve? How can the equation of the tangent line be the same equation throughout the curve?

Answer: 
The derivative of a function at a point is the *slope* of the tangent line at that point, not the line's equation. The function $f'(x)$ tells you this slope for each point in the domain of $f$ where a non-vertical tangent line exists.
**Hint**

Suppose that you run a car and that you registered the distance as a function of time; this means that you have a function $$\text{distance}=f(\text{time})$$ The derivative is the change of the distance over a small period of time; this is the speed $$\text{speed}=f'(\text{time})$$ When considering a curve, the derivative of the function at a given value of $x$ gives the slope of the tangent line to the curve at this specific point.