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$$\frac{x^3}{9}\Bigg|_0^1$$ The vertical line above: what does it mean, and how would I state this whole structure in spoken words, so that a screen reader would be able to read it aloud correctly?

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    Evaluate. In particular here, $f(x)|_a^b$ usually means $f(b)-f(a)$. – 2017-01-04

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It looks like people have already told you what to do mathematically. I'm a math tutor, and what I say out loud is

"x cubed over nine, evaluated from one to zero."

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    Yes, of course! Thank you; an edit on the way! – 2017-01-04
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    Wait, why would you say from 1 to 0 and not vice versa? I normally say from 0 to 1 because we go from left to right. I think saying from 1 to 0 we should pick up a minus sign. – 2017-01-04
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    That's interesting. How I learned it was with the bar like this we say "evaluated from 'upper bound' to 'lower bound'". However, when we integrate we say "an integral from 'lower bound' to 'upper bound'". Seems like 'evaluated' is the key word. Not saying that this is the only 'right' way, though. Really, it's just about what people around you and your professors said most. – 2017-01-04
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The vertical line means evaluate it from the top to the bottom.

So say $x^3/9$ from $0$ to $1$.

in general $f(x)\biggr \vert^b_a$ would be $f(b)-f(a)$, in this case:

You would evaluate it as $(1)^3/9 - (0)^3/9$.

Basically just evaluate the expression with $x=$ top limit and the bottom limit, subtract the bottom expression from the top expression.

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    How would I write this out in spoken words, so a screen reader would state it correctly aloud? Thank you. – 2017-01-04
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    @Chelonian This is context vital to the question; it sounds like you're working on accessibility, which, if true, is quite the lofty and commendable goal! – 2017-01-04
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    @pjs36 Yes, thank you. – 2017-01-04
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    Wait, why would you say from 1 to 0 and not vice versa? I normally say from 0 to 1 because we go from left to right. I think saying from 1 to 0 we should pick up a minus sign. – 2017-01-04
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It is the standard notation for describing the limits of an integral.

As well mentioned in comments and other answers, $f(x)\biggr \vert^b_a$ is same as $f(b)-f(a)$ and here $a$ and $b$ are called limits of the integral or the integral is said to be over the interval $[a,b]$ .

When you write something like $\frac{x^3}{9}\Bigg|_0^1$ you are gonna say it as

Evaluate $\frac{x^3}{9}$ over the interval $[0,1]$ or evaluate $\frac{1^3}{9}-\frac{0^3}{9}$.