In this definition, do the subsequences always have an infinite number of terms? I'm a bit confused about the notations used here. Are $\{n_k\}$ the sequence of ALL positive integers, i.e. $\{n_k\} = \{1,2,3,4,5,6,...\}$, and what the definition says is that we pick an infinite subset of $\{n_k\}$, and use them as indices in $\{{p_n}_i\}$?
Yes, they always have an infinite number of terms, that is implied by the dots in $n_1 < n_2 < n_3 < \cdots $
$\{n_k\}_{k=1}^{\infty}$ can be any infinite sequence of natural numbers. For example it might be the case that you want $n_k = 2k$, or $n_k = k^2$ for all $k\in\mathbb{N}$.
So, if you're original sequence equals $$ (1,4,9,16,25,36,49,64,81,\dots) $$ then the subsequence in the first example equals $$ (4,16,36,64,\dots) $$ and in the second example equals $$ (1,16,81,256,\dots) $$
I hope this helps. If you have any questions feel free to comment!