A useful heuristic when thinking about projective systems is that you want to specify a congruence condition modulo lots of numbers (or ideals) in a compatible way. For example, you can specify any congruence condition modulo 3 and any congruence condition modulo 5. No matter, what you specify, the two can be satisfied at the same time, since 3 and 5 are co-prime: this is the Chinese remainder theorem. But if you specify a congruence condition modulo 5 and modulo 25, then the two could turn out to be incompatible. E.g. there is no number that is 0 modulo 5, but 1 modulo 25. Why? Because a number that is 0 modulo 5, is divisible by 5, so it can only by congruent to 0,5,10,15, or 20 modulo 25. Similarly, once you have chosen a congruence class modulo 3 and modulo 5, the congruence class modulo 15 is uniquely determined, while the congruence class modulo 30 is not unique but heavily restricted.
The projective limit of $\mathbb{Z}/n\mathbb{Z}$ that you have described is the set of all legal simultaneous choices of congruence classes modulo all integers (with the obvious ring structure). The projective system $\mathbb{Z}/m\mathbb{Z}\rightarrow \mathbb{Z}/n\mathbb{Z}$ for $n|m$ expresses all the above rules, that I have spelled out for 3 and 5.
Similarly, if you fix a prime $p$, then you can consider the projective system $\mathbb{Z}/p^m\mathbb{Z}\rightarrow \mathbb{Z}/p^n\mathbb{Z}$ for $n\leq m$. The projective limit is denoted by $\mathbb{Z}_p$ and is the set of all possible legal simultaneous choices of congruence classes modulo higher and higher powers of $p$. Thus, the above example shows that $(0,5,\ldots)$ could represent an element of the projective limit, while $(0,1,\ldots)$ cannot.