Q:

What does it mean for a set to be "nested"?

What does it mean for a set  to be "nested"?
and can you please show an example of that is and one that isn't

A:

Since you have tagged you question real analysis, you are probably interested in nested sequences of sets, which appear, for example, in Cantor intersection theorem.
We call a sequence $(A_n)_{n=1}^\infty$ of sets a nested sequence of sets if the next set is always a subset of its predecessor, i.e.,
$$(\forall n\in\{1,2,\dots\}) A_{n+1} \subseteq A_n.$$
So the nested sequence looks like this $A_1 \supseteq A_2 \supseteq \dots \supseteq A_n \supseteq \dots$
See also Wikipedia article about nested intervals.
So examples of nested sequences of subsets of $\mathbb R$ would be:

$A_n=(-\frac1n,\frac1n)$
$A_n=[n,\infty)$
$A_n=[0,1+\frac1n)$

If we put, for example, $A_n=\{n\}$, then these sets are not nested.

According to Wikipedia, there exists a notion of nested set used in set theory. I was unaware about this notion and I don't know anything about it - so except for the link to Wikipedia article I can't give you more information. But this is probably not what you were after.

