On wikipedia, the normal crossing divisor is defined to be (by my understanding):


(Assume $X/k$ be a smooth geometrically integral scheme of finite type over a field $k$).


Let $D = \sum\_{i=1}^n C\_i$ be a Weil Divisor, here $C\_i$ are irreducible closed subsets of codimension 1 of $X$. Endow $C\_i$ with the reduced scheme structure (hence they are integral closed $k$-scheme of $X$ of codimension 1.) We call $D$ is a normal crossing divisor if each $D\_i$ is smooth over $k$ and $D\_i$'s intersect transversely.


But in somewhere, I saw the notion "strict normal crossing divisor". For example,


Definition 1.5.1, p.8, in <http://math.arizona.edu/~swc/aws/07/KedlayaNotes10Mar.pdf>


or


5.1, p.16, in <http://www.uni-due.de/~bm0032/publ/TubNbdDocumenta.pdf>


The definition looks the same as "normal crossing divisor".


I would like to know what's the difference between these two definitions. It 
would be great with examples. Also, if the base scheme $S$ is not a field, then is the definition the same as above with the modification "each $D\_i$ is smooth over $S$"?



normal crossing divisor v.s. strict normal crossing divisor
Your definition of normal crossings divisor is, as you say, often called a strict normal crossings divisor. People who use this terminology allow normal crossings divisors to have components which are not necessarily smooth; they are only required to looks like a smooth components meeting transversally locally in the analytic topology (if one works over $\mathbb{C}$) or etale locally (in general).


As an example, an irreducible curve in a smooth surface having (only) an ordinary node as a singularity would be called a normal crossing divisor but would not be a strict normal crossings divisor. 


