The slope of a line is a measure of how steep the line is. If you go along a line and as a result you moved a distance \(\Delta x\) across and a distance \(\Delta y\) up, then the slope of the line is given by:
$$ \text{slope} = \frac{\text{distance up}}{\text{distance across}} = \frac{\Delta y}{\Delta x} $$
Use the interactive visual below to see how the slope of the line connecting the two yellow dots changes as you drag them around. The visual area is \(5\) units wide and \(5\) units high.
In case you didn't noticed it, the delta symbol \(\Delta\) is the Greek letter "D" and it's used to denote a change in a quantity. So:
$$ \begin{align*} \Delta x &= \text{change in } x \\ \Delta y &= \text{change in } y \end{align*} $$
If the dots are arranged such that \(\Delta x = 2\) and \(\Delta y = 4\), then the slope is \(2\). This means that for every unit you move to the right, you move \(2\) units up. In other words, the rate of change of \(y\) with respect to \(x\) is \(2\) for a line with a slope of \(2\). Note that you can move any distance to the right and the change in \(y\) will be \(2\) times the change in \(x\). Therefore the slope (or the rate of change) act as a multiplier of the change in \(x\). If you know the slope of a line, you can calculate the change in \(y\) for any change in \(x\). The slope converts the change in \(x\) to a change in \(y\):
$$ \begin{align} \text{slope} \times \Delta x = \Delta y \end{align} $$
Now, this is all easy and nice if the dots are connected with a STRAIGHT LINE. This is what we call a linear function or relationship between \(y\) and \(x\). What if the way of getting from \(x\) to \(y\) is more complicated?
The visual above shows a more complicated relationship between \(x\) and \(y\). The path connecting the yellow points is not a straight line. You can change the shape of the path by dragging the yellow and orange points. The slope between any two points can now be different. But what does the slope mean in this case? In the previous example, the slope was the rate of change of \(y\) with respect to \(x\). You could move any distance to the right and the change in \(y\) would be the slope times the change in \(x\). Now, the slope between two points is the average rate of change of \(y\) with respect to \(x\) between the two points. You still know how much \(y\) changed for a given change in \(x\), but you don't know anything about the rate of change at any point in between the two points. You couldn't reconstruct the path any better than saying it was a straight line! Suddenly, the idea of a slope at a single point does make sense. By bringing the blue points closer together, you could calculate the slope at a point on the path. This is the idea behind the derivative in calculus. Choosing two points that are very close together, the slope between them is a good approximation of the slope at a single point. Then you could use again equation \((1)\) to calculate the change in \(y\) for a very small step forward in \(x\):
$$ \begin{align*} \text{slope} \times \Delta x \approx \Delta y \end{align*} $$
Amazing things happen when you start to abstract this to physical quantities like the position of an object with respect to time. If you go in a straight line with a constant speed, your position vs. time path is a straight line and the slope is the speed. If you then start to accelerate, the path becomes a curve and then the slope between two points in time is the average speed between the two points. The slope at a single moment in time is the speef at that moment.