Consider the following statement:
If $X$ is partially ordered set such that every chain in $X$ has un upper bound, then for every $x \in X$ there is a maximal element $m$ in $X$ such that $x \le m$.
What is the relation, if any, to Zorn's lemma? Is it weaker, stronger or maybe just nonsense?
The statement is equivalent to Zorn's lemma.
It implies Zorn's lemma quite easily, because they have the same requirements from the partial order, and if there is a maximal element above each point, then there's certainly a maximal element.
On the other hand, assuming Zorn's lemma, and given a partial order $(P,\leq)$ satisfying these requirement, consider for $x$ in the partial order the set $P_x=\{m\in P\mid x\leq m\}$, then the restriction of $\leq$ to $P_x$ satisfies Zorn's lemma again, and therefore it has a maximal element, $m$ which is maximal in $P$, and so $x\leq m$.