How does the $\mathcal Z$-transform's "region of convergence" work?
To be completely honest, I thought the theory behind the Z-transform was kinda opaque in college too. In hindsight, taking a course in complex analysis would have made it clearer. And I too dislike the notational conventions that seem to be used for this stuff. Strictly speaking, the usual convention here is that


$x[n]$ denotes a discrete-time sequence

$n\in \mathbb{Z}$
the brackets denote a discrete argument

$X(z)$ denotes a continuous-valued transformed function

$z\in\mathbb{C}$ (is a complex number)
the parentheses denote a function accepting a continuous-valued parameter
the capital $X$ denotes a transformed version of some other function/sequence $x$ (a similar notation is used for Fourier transforms: $F(j\omega)\leftrightarrow f(t)$




  What do they mean by z=0? Are they substituting z as 0, if so in which equation?


They mean, just plug $z=0$ into your usual definition of the Z-transform.

$X(z) = \sum_{n=\infty}^{\infty} x[n] z^{-n}$

Generally (more precisely, when $x[n] \ne 0$ for some $n \ne 0$), this sum will diverge (to infinity) for some complex $z$. For instance, let $x[0] = 1, x[1] = 1$, and $x[n]=0$ for $n&lt;0$ and $n&gt;1$. Then $X(z) = 1 + z^{-1}$. The ROC does not includes $z=0$, for $\lim_{z\rightarrow 0} X(z)=\infty$

When your text says "When $n_2&gt;0$ there will be a $z^{−1}$ term and thus the ROC will not include $z=0$", what they mean by that is, when $x[n]$ is nonzero for some $n&gt;0$, it is unavoidable for the z-transform to include a $z^{-n}$ term, which diverges to infinity at $z=0$. That's all.


  How do we calculate the region of convergence for an infinite sequence?


Lots of math. Ha!

srsly, the way this is done is to obtain an algebraic formulation for the sequence in question, plug it into the Z-transform definition, and use the tools available from the analysis of geometric series (and complex power series) to determine where this Z-transform converges/diverges. In practice, determining whether $|z|=1$ converges is the most important question to answer, because that determines stability, and whether or not you can obtain a frequency response from the system, etc. But causality might matter too, depending on what you're doing.