The slope of a line is "rise over run" as I was taught - that is, how much it rises over a certain distance of horizontal distance. This definition assumes that we're visualizing the function in graph or plot that places the domain of the function, x, corresponding to the horizontal axis (the "run")and the range of the function, y, corresponding to the vertical axis (the "rise"). So the ratio of the rise to the run is the slope.
In a function like x/8-1/4 as the value of y, you can discount the 1/4 because it has no impact on the rate of the rise because there's multiplier for x. In other words, no matter what value of x you plug in, the impact of the "-1/4" is the same - it just moves the graph downwards a uniform amount for all values of x.
The key part is y=x/8. This says that the value of y is one eighth to value of x. So if you add 1 to x you are only adding 1/8 to y. Add 2 to x, you add 2/8 or 1/4 to y.
So the answer to your question is 1/8. It is the answer to an equivalent question of "by how much does y increase when x increases by 1?"