The standard example, often taken to be the default, is that of morphisms in the category sSet of simplicial sets which have the left lifting property against all Kan fibrations. In this case, anodyne morphisms are equivalent to acyclic cofibration in the standard model structure on simplicial sets.
Typically one says anodyne morphism or, actually, anodyne extension if one thinks of these morphisms produced by retracts and transfinite composition of pushouts of a certain generating set of morphisms.
So in the standard example of left lifting against Kan fibrations, one typically speaks of anodyne extension if one produces morphisms by these operations from the set of horn inclusions. (see for instance (Jardine)).
So is anodyne if for every Kan fibration and every commuting diagram
there exists a lift
See for instance (Jardine) for details.
Similarly a morphism is called
See (Lurie) (following Joyal).
In the category of dendroidal sets there is a notion of horn inclusions that generazies that of simplicial sets. The corresponding saturated class of morphisms is called that of dendroidal inner anodyne morphisms.
See (Cisinski-Moerdijk 09).
Kan fibration, anodyne morphism
Left/right and inner anodyne morphisms of simplicial sets are discussed in section 2 of
Inner anodyne morphisms of dendroidal sets are discussed in