A vertical asymptote does mean undefined in a way, but asymptotes act like limits to a function, meaning a function will never intercept them.
For example:
Take the function: f(x)=1/x^2-9
A vertical asymptote occurs when the domain cannot be a certain value, thus the function cannot cross the line for that value because there is no input nor output. So, in a way, they can be considered undefined.
In the example, we must set the denominator equal to 0 in the rational function, since a number divided by 0 is undefined. We are trying to figure out what value for x creates a denominator of 0, and that will be a vertical asymptote.
x^2-9 is the denominator terms
x^2-9=0
We must solve for x to figure out what domain value for x equals a denominator of 0:
Add 9 to both sides:
x^2=9
Take the positive and negative square roots of 9:
±√x^2=±√9
x=3, -3
So, our vertical asymptotes in this example are the lines x=3 and x=-3
This is because 1/(3)^2-9=1/0 and 1/(-3)^2-9=1/0, both of which are undefined.