If $A$ is a local  ring, its henselization $i:A\to A^h$ is always injective and even faithfully flat.  

The rings $A$ and $A^h$ have the same dimension and share many properties :   
$A$ is noetherian (resp. reduced, resp. a normal domain) $\iff$ $A^h$ is noetherian (resp. reduced, resp. a normal domain).     

And if $A$ (and thus $A^h$ ) is noetherian we can add that $depth(A)= depth(A^h)$ and that :  
$A$ is regular  (resp. Cohen-Macaulay) $\iff $ $A^h$ is regular  (resp. Cohen-Macaulay).  

Notice that a complete local ring is henselian and that in general a local ring and its completion have the same completion : the morphism $\hat A \stackrel {\cong}{\to} \hat {A^h}$ is an isomorphism.  

**Edit: Bibliography**   
The main source is  (surprise, surprise! ) EGA IV,4,§18 which contains proofs of most of the above.  

Raynaud wrote a rather elementary  (scheme theory not assumed) self-contained monograph [*Anneaux Locaux Henséliens* ](http://books.google.fr/books/about/Anneaux_locaux_Hens%C3%A9liens.html?hl=fr&id=5gYvAAAAIAAJ).     
 It is short (129 pages) but packed with highly non trivial results like a complete proof of Zariski's  Main theorem.  
I felt  a feeling  of loss and nostalgy when reading in the Introduction that Raynaud thanked  N. Bourbaki for having allowed him  access to his [Bourbaki's] book "in preparation" (!) on étale algebras ...    

For a recent  presentation, one can look at Section 38 of De Jong and collaborators' ever growing (as of today: 3150 pages  =24.4 Raynauds ) [Stacks Project](http://www.math.columbia.edu/algebraic_geometry/stacks-git/book.pdf)