A Henselian ring is a local ring for which the conclusion of Hensel's lemma? (classically stated for a complete local ring?) holds.
Let $R$ be a local ring with maximal ideal $m$. Let $k = R/m$ be the residue class field, and for a polynomial $f \in R[x]$, let $\bar{f} \in k[x]$ obtained by reduction of the coefficients of $f$ modulo $m$. Then $R$ is Henselian if, for any monic $f \in R[x]$ such that $\bar{f}$ factorizes as $g_0 h_0$ with $g_0, h_0$ monic and relatively prime in $k[x]$, there exist monic $g, h \in R[x]$ such that $f = g h$ with $\bar{g} = g_0$, $\bar{h} = h_0$, and the ideal $(g, h)$ is the unit ideal in $R[x]$.
Often this definition is given just for the case when one of $g_0$, $h_0$ is a linear factor $x - a$, where the idea is that the (simple) root $a$ can be lifted to $R$.
A Henselian ring $R$ with residue class field $k$ is strictly Henselian if $k$ is separably closed.
Any field $k$ is trivially Henselian.
A complete local ring, such as p-adic number rings, is Henselian. This includes rings of formal power series over a field, $k[ [x_1, \ldots, x_n] ]$.
Rings of convergent power series over a local field are Henselian.
The quotient of a Henselian ring is also Henselian.
In the first place, a quotient $R'$ of a local ring $R$ is local (and has the same residue class field $k$). Secondly, $R'$ is clearly Henselian since we can lift factorizations $\bar{f} = g_0 h_0$ in $k[x]$ to factorizations in $R[x]$, and then push them down to $R'[x]$.
If $R$ is local and its reduced ring (i.e., $R$ modulo its ideal of nilpotent elements) is Henselian, then $R$ itself is Henselian.
If $R$ is Henselian and $S$ is a local ring that is integral over $R$ (meaning that $S$ is an $R$-algebra and each $x \in S$ is an integral element over $R$), then $S$ is Henselian.
Let $LocRing$ denote the category of local rings (commutative of course) and local ring homomorphisms ($f: R \to S$ is local if the pullback along $f$ of the maximal ideal of $S$ is the maximal ideal of $R$).
The full subcategory of $LocRing$ consisting of Henselian rings is reflective. The left adjoint to the full inclusion is called Henselization.
Let $R$ be a discrete valuation ring, with $K$ its field of fractions. Let $\hat{R}$ be the $m$-adic completion of $R$ with respect to its maximal ideal. Then the Henselization of $R$ is isomorphic to the subring of $\hat{R}$ whose elements are roots of separable polynomials with coefficients in $K$.
A rule of thumb, as suggested by this example, is that the Henselization is the algebraic part of a local ring completion.
An original reference is
Lecture notes are in
Other sources include
Jacob Lurie, Descent Theorems, section 3.
Alonso, Lombardi, Perdry, Henselian local rings (pdf)
Krzysztof Jan Nowak, Remarks on Henselian rings (pdf);
Ieke Moerdijk, Rings of smooth functions and their localizations (pdf)