If $R$ is a nonzero commutative unitary ring, then $R$ has a maximal ideal. Indeed, every proper ideal $I$ in $R$ is contained in a maximal ideal.
There is a proof of this in Rotman's Advanced Modern Algebra, but there is something that I do not get in the second paragraph...
In the second paragraph of the proof, it says "Let $X$ be the family of all proper ideals containing I partially ordered by inclusion...". I'm not sure why we can assume that inclusion gives us a partial order. If we look at the ring $\Bbb{Z}$, for instance, we see that the ideal $(0)$ is contained in both $(2)$ and $(3)$. However, it is not true that $(2) \subseteq (3)$ or that $(3) \subseteq (2)$.
A partial order needn't be total, there are, as you noted, incomparable elements. Recall that a partial order on a set $X$ is an antisymmetric, reflexive and transitive relation $R\subseteq X\times X$.