In Nielsen and Chuang, when talking about the Deutsch–Jozsa algorithm. The Deutsch’s problem is described as the following game.
Alice, in Amsterdam, selects a number x from 0 to $2^n − 1$, and mails it in a letter to Bob, in Boston...
Is there an underlying reason why we use numbers from 0 to $2^n − 1$ or it's just a random choice?
Numbers from $0$ to $2^n-1$ are exactly what one can encode into $n$ bits or $n$ qubits using standard binary positional number system. The mapping looks like this
$$ \begin{align} &0\dots000 \mapsto 0 \\ &0\dots001 \mapsto 1 \\ &0\dots010 \mapsto 2 \\ &0\dots011 \mapsto 3 \\ &0\dots100 \mapsto 4 \\ &0\dots101 \mapsto 5 \\ &\ \ \dots \\ &1\dots111 \mapsto 2^n-1. \end{align} $$
In other words, it follows the same pattern as the well-known decimal notation only using two symbols instead of ten. Note that $n$ decimal positions correspond to numbers from $0$ to $10^n-1$.
