Roughly speaking, Hensel's Lemma states that a polynomial $f \in O[X]$ over a certain local ring $(O,\mathfrak{m})$ which factors over the residue field $O/\mathfrak{m}$ into coprime polynomials also factors over $O$ in a compatible way. However, there are different versions of the lemma with different requirements on $O$. For example the statement holds true if $O$ is complete with respect to the $\mathfrak{m}$-adic topology, and it also holds true if $O$ is the valuation ring of a nontrivial non-archimedean absolute value on some field $K$, which is complete with respect to this absolute value.
Is there a more general version of Hensel's Lemma which implies both of the above statements?
Let $(R, \mathfrak{m})$ be a local ring, $k = R/\mathfrak{m}$ its residue field, $S = \operatorname{Spec}R$, and $s$ the closed point of $S$. The following conditions are equivalent.
$(i)$ Every finite $R$-algebra $A$ is a direct product of local rings.
$(ii)$ The condition $(i)$ holds for $A = R[t]/(f (t))$ for any monic polynomial $f (t) \in R[t]$.
$(iii)$ For any finite $R$-algebra $A$, the canonical homomorphism $A \rightarrow A/\mathfrak{m}A$ induces a one-to-one correspondence between the set of idempotent elements in $A$ and the set of idempotent elements in $A/\mathfrak{m}A.$
$(iv)$ The condition $(iii)$ holds for $A = R[t]/(f (t))$ for any monic polynomial $f (t) \in R[t].$
$(v)$ For any monic polynomial $f (t) \in R[t]$ and any factorization $\overline{f}(t) = \overline{g}(t) \overline{h}(t)$, where $\overline{f}(t)$ is the image of $f (t)$ in $k[t]$, and $\overline{g}(t)$ and $\overline{h}(t)$ are relatively prime monic polynomials in $k[t]$, there exist uniquely determined polynomials $g(t)$ and $h(t)$ in $R[t]$ such that $f (t) = g(t)h(t)$, $\overline{g}(t)$ and $\overline{h}(t)$ are images of $g(t)$ and $h(t)$ in $k[t]$, respectively, and the ideal generated by $g(t)$ and $h(t)$ is $R[t].$
$(vi)$ The condition $(v)$ holds for any factorization $\overline{f} (t) = (t− \overline{a}) \overline{h}(t)$ such that $t − \overline{a}$ and $\overline{h}(t)$ are relatively prime monic polynomials in $k[t].$
$(vii)$ For any etale morphism $g : X \rightarrow S$, any section of $g_s : X \otimes _R k \rightarrow \operatorname{Spec} k$ is induced by a section $g.$
If a local ring $(R, \mathfrak{m})$ satisfies any (and hence every) of these properties, we call the ring henselian.
Examples: (1) A complete local ring is henselian.
(2) The convergent power series ring $\mathbb{C}\{z_1, z_2, \dots , z_n\}$ is also henselian. (This ring is not a complete local ring.)
The above definition is taken from Etale Cohomology Theory, Lei Fu. You can also look at here.