When we say that $S$ is a poset, we usually are already referring to a specific order relation, that is, we actually mean $\langle S, \leq \rangle$ is a poset, where $\leq$ is any binary relation defined on $S$ which is reflexive, anti-symmetric and transitive.
So the partial ordering is often understood from the context, or from a previous reference to it.
The user JMoravitz gives in a comment a good example of two simple partial orderings on $\mathbb N$.
The idea that we mean that there can be defined a partial ordering on that set, I think it doesn't make much sense, because it is vacuous: for any set $S$, we can always define the poset $\langle S,=\rangle$.
It's an anti-chain, but a poset nonetheless.
It is also always possible to define a chain in any set (just take the anti-chain above, or any other ordering, and apply Szpilrajn Theorem).
When we say that a subset of $S$ has a least upper bound (or greatest lower bound) we are, of course, doing it with reference to the understood partial ordering.