Calculus is all to do with variation. The steepness of the curve or straight line is really just how it varies at different points.
This is why calculus is used to much in physics. Much of physics is all about variation, and so calculus is just a neat way of describing it (maths is just a language, and it's the most exact language for describing what we want to describe).
One example is distance. I'm sure you're familiar with the concept of distance. But what if you want to see how the distance varies as time goes by? Often we want to do this, and we call it speed. Again, something I'm sure you're familiar with. So if you're in a car and move at the exact speed all the time, you could draw the distance you make as time goes by on a graph, and you'd have a straight line. The steepness of the line is what we call speed, and since speed is constant in this example, the steepness is constant, and so it's a straight line.
But what if the speed is not constant? I mean, usually it isn't. Usually if you're sat waiting for the traffic lights to change, your speed will be zero, and then you're slowly increase your speed. So this graph that was a straight line is now going to be a curved line, as the steepness (the speed, remember) isn't the same at every point you've driven. So to find the steepness of the curve at a certain point (the speed you were going at that point), you need calculus (specifically differentiation). It's just a tool you use to find the steepness, which is representing the speed.