Henselian local ring

Symbol-free definition
Let $$R$$ be a local ring with maximal ideal $$I$$. We say that $$R$$ is Henselian if given any polynomial $$f(x) \in R[x]$$ and $$a$$ such that:

$$f(a) \equiv 0 (\mod f'(a)^2I)$$

we can then find a $$b$$ such that:

$$f(b) = 0$$ and $$b \equiv a (\mod f'(a)I)$$

Another way of putting it is that the ring must satisfy Hensel's lemma with respect to its unique maximal ideal.