What does it mean that numbers are "totally ordered", in the Paxos Made Simple paper?
Totally ordered means that there exists an order relation such that given any two elements, one is greater than the other. In other words, there are no incomparable elements. For example, the real numbers are totally ordered: either $x \le y$ or $y \le x$. Counter-example: set inclusion $\subseteq$ is an order relation, but $\{0\}$ and $\{1\}$ are incomparable.

Intuitively, a total order is one where you can lay all the elements on a line, where moving along the line always increases the value. (This is not mathematically true because some orders are too “dense” to fit.)

Suppose that two proposals numbered $n_1$ and $n_2$ are chosen, with values $v_1$ and $v_2$ respectively. The desired safety property is that the values are in fact the same: $v_1 = v_2$. By condition P2, if $n_1 \le n_2$, then proposal $n_2$ is higher-numbered than proposal $n_1$, so its value $v_2$ is equal to $v_1$. Similarly, if $n_2 \le n_1$ then $v_1 = v_2$. The fact that the integers are totally ordered implies that at least one of the propositions $n_1 \le n_2$ and $n_2 \le n_1$ is true, and therefore either way $v_1 = v_2$.

What condition P2 means in some idealized way is that if a choice is made, all subsequent choices will be for the same value. However, “subsequent” doesn't quite work out, because the numbering of the proposals doesn't necessarily correspond with the order in which they are made in real time: different nodes in the network can make proposals before they are aware of each other, so numbers can't be assigned to proposals in chronological order. Solving this problem is a large part of working out a consensus algorithm.