In Awodey's Category Theory, page 130, he says:
A poset is (co)-complete if it is so as a category, thus if it has all set-indexed meets (resp. joins). For posets, completeness and cocompleteness are equivalent. A lattice, Heyting algebra, Boolean algebra, etc. is called complete if it is so as a poset.
As an example, I'll take the poset $\omega$, which is well-ordered. A subset of a well-ordered set has a least element, so $\omega$ has all meets.
However, $\omega$ itself (as a subset of $\omega$) has no join. Indeed, a join would be $a$ such that for all $c \in \omega$, have "all $a_i \leq c$" iff $a \leq c$. Since there is no $c \in \omega$ such that for all $a_i \in \omega$, $a_i \leq c$, we must therefore have $a \not \leq c$ for all $c \in \omega$. But no $a$ has this property, so $\omega$ doesn't have a join.
In summary, $\omega$ has all meets, but doesn't have all joins. That is, it is complete but not cocomplete.
What have I done wrong? Awodey makes the distinction between a Boolean algebra and a poset here, so we don't necessarily have access to the nice properties of Boolean algebras.
In a preordered set $(X,\le)$, a maximum element (which is the same thing as a terminal object in the associated category) can be described either as the infimum of the empty system or as the supremum of all elements of $X$. This shows that $\omega$, which lacks a maximum element, is neither complete nor cocomplete as a category.
This is due to the observation, that when checking universal properties in thin categories, uniqueness of the induced arrows is for free. That is, a terminal object $t$ is characterized by the existence of some arrow $x\longrightarrow t$ for all objects $x$. Both, $t$ being the product of the empty family, and $t$ being the coproduct of all objects, ensure this condition.