From Rudin's Principles of mathematical analysis,
6.2 Definition
Let $\alpha$ be a monotonically increasing function on $[a,b]$. ... Corresponding to each partition $P$ of $[a,b]$, we write
$$\Delta \alpha_i = \alpha(x_i) - \alpha(x_{i-1}).$$
He then goes on to define the Riemann Stieltjes integral of $f$ with respect to $\alpha$, over the interval $[a,b]$.
The Riemann integral is then pointed out to be a special case of this when $\alpha(x)=x$.
With $\alpha(x)=x$, I understand $\Delta x = x_i - x_{i-1}$ to represent the directed magnitude of the "base of the approximating rectangle" that we then multiply by the value of $f$ taken somewhere within this interval, thus obtaining the area of an approximating rectangle.
I don't know where to begin to interpret the case where $\alpha(x) \not\equiv x$.