Let $(P,\le )$ be a partially ordered set. We call a $m\in P$ a maximal element, if $$p\in P,m\le p\Rightarrow m=p$$
What is the negation of this property? So we assume $m\in P$ is not a maximal element. I would say the following: There exists $p\in P,m\le p \nRightarrow m=p$, i.e. $m< p$. Is this correct?
Not entirely, no: you simply want to say that there is an element of $P$ that is greater than $m$. That’s simply $\exists p\in P(m<p)$ or, if you haven’t defined $<$ yet, $\exists p\in P(m\le p\land m\ne p)$.
The point is that the negation of $m\le p\to m=p$ is $m\le p\land m\ne p$: more generally, $\varphi\to\psi$ is equivalent to $\neg\varphi\lor\psi$, whose negation is $\neg(\neg\varphi\lor\psi)\equiv\varphi\land\neg\psi$ by one of the De Morgan laws.