In ZF, many of the usual arguments about critical points still go through.

For example, every critical point $\kappa$ of an elementary embedding $j:V\to M$ is regular, since if $\kappa$ is the supremum of a short sequence $s$ below $\kappa$, then it is easy to see that $j(s)=s$, which would imply $j(\kappa)=\kappa$, contradicting the assumption that $\kappa$ is the critical point.

If $\kappa$ is the critical point of $j:V\to M$, then $\kappa$ is weakly inaccessible. To see this, note that if $\kappa=\delta^+$, then $j(\kappa)=(\delta^+)^M$, which of course is at most $\kappa$, since $P(\delta)^V=P(\delta)^M$, and this contradicts $\kappa<j(\kappa)$. So $\kappa$ is a regular limit cardinal.

If $\kappa$ is the critical point of $j:V\to M$, then $\kappa$ is a measurable cardinal. To see this, let $\mu$ be the set of $X\subset\kappa$ for which $\kappa\in j(X)$, and the usual arguments show that this is a $\kappa$-complete nonprincipal ultrafilter on $\kappa$, without using the axiom of choice. So $\kappa$ is measurable. (Note: in ZF measurability is weaker than you might expect, since even $\omega_1$ can be measurable, so it doesn't imply inaccessibility.)

If $\kappa$ is the critical point of $j:V\to V$, then $\kappa$ is $<j(\kappa)$-supercompact, in the sense that there is a normal fine measure on $P_\kappa\lambda$ for every $\lambda$ in the interval $[\kappa,j(\kappa))$, because you can let $\nu$ be the set of $X\subset P_\kappa\lambda$ with $j"\lambda\in j(X)$, and the usual arguments show that this is a normal fine measure. (Note, without AC, the equivalence formulations of supercompactness break down, since one can't always establish that the ultrapowers are elementary.)

Similarly, if $\kappa$ is the critical point of $j:V\to V$, then $\kappa$ is (weakly) extendible, since for every $\lambda>\kappa$ we may look at $j\upharpoonright V_\lambda:V_\lambda\to V_{j(\lambda)}$, which is elementary and verifies weak extendibility (weak = we do not insist on $\lambda<j(\kappa)$, which is equivalent in ZFC using the Kunen inconsistency).